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/■ary Sytttm
ity of WisconsinMaditoil
otate Street
dison, Wl 537061484
S.A.
A MANUAL
OF
RULES, TABLES, AND DAT4
FOR
MECHANICA^ lENGINEERS
BASED ON THE MOST RECENT INVESTIGATIONS:
OF CONSTANT USE
IN CALCULATIONS AND ESTIMATES RELATING TO
Stkskcth of Matbkials AMD OP Elementary Constructions; Labour;
Heat and its Applications, Steam and its Properties, Combustion and Fuels,
Steam Boilers, Steam Engines, HotAir Engines, GasEngines ; Flow op Air and op
Water; Air Machines; Hydraulic Machines; Millgearing; Friction and the Resistance of
m achinbry, &c ; weights, measures, and monies, british and forbign, with the reciprocal
Equivalents for the Conversion of British and French Compound Units op
Weight, Pressure, Time, Space, and Moi^BV : Specific Gravity and ^
THE Weight op Bodies; Weight op Metals, &c.
with
Tables of Logarithms, Circles, Squares, Cubes, SquareRoots, and CubeRoots;
AND MANY OTHER USEFUL MATHEMATICAL TaBLBS.
BY
DANIEL KINNEAR CLARK,
% I
AUTHOR OF
MEMBER OF THE INSTITUTION OP CIVIL ENGINEERS;
RAILWAY MACHINERY," "EXHIBITED MACHINERY OF 1862," BTC.
SECOND EDITION.
* LONDOVsT:
BLACKIE & SON: PATERNOSTER BUILDINGS,
GLASGOW AND EDINBURGH.
1878.
A/i Rights Reset ved.
QhABQOWl
W. O. BLAOKIS AJSD CO., PBINTEBS,
VILLAFIXLD.
PREFACE.
This Work is designed as a book of general reference for Engineers :
— ^to give within a moderate compass the leading rules and data,
with numerous tables, of constant use in calculations and estimates
relating to Practical Mechanics. The Author has endeavoured to
concentrate the results of the latest investigations of others as well
as his own, and to present the best information, with perspicuity,
conciseness, and scientific accuracy.
Amongst the new and original features of this Work, the follow
ing may be named : —
In the section on Weights and Measures, the weight, volume,
and relations of water and air as standards of measure, are concisely
set forth. The various English measures, abstract and technical,
are given in full detail, with tables of various wiregauges in use:
and equivalent values of compound units of weight, power, and
measure — as, for example, miles per hour and feet per second.
The French Metric Standards are defined, according to the latest
determinations, with tables of metric weights and measures, equi
valents of British and* French weights and measures, and a
number of convenient approximate equivalents. There is, in addi
tion, a full table of equivalents of French and English compound
units of weight, pressure, time, space, and money — as, for example,
pounds per yard and kilogrammes per metre; which will be found
of great utility for the reciprocal conversion of English and French
units.
The tables of the Weight of bars, tubes, pipes, cylinders, plates,
sheets, wires, &c., of iron and other metals, have been calculated
expressly for this Work, and they contain several new features
designed to add to their usefulness. They are accompanied by a
summary of the various units of weight of wrought iron, cast iron,
and steel, with plain rules for the weight.
In the section on Heat and its Applications, the received mechan
ical theory is defined and illustrated by examples. The relations
of the pressure, volume, and temperature of air and other gases.
vi PRfeFACE.
with their specific heat, are investigated in detail. The transmission
of heat through plates and pipes, between water and water, steam
and air, &c., for purposes of heating or cooling, is verified by many
experimental data, which are reduced to units of performance.
The physical properties of steam are deduced from the results
of Regnault's experiments, with the aid of the mechanical theory
of heat A very full table of Ihe Properties of Saturated Steam is
given. The table is, for the most part, reproduced from the article
** Steam," contributed by the Author to the Encyclopedia Britannicay
8th edition, and it was the first published table of the same extent,
in the English language, based on Regnault's data. An original
table of the properties of saturated mixtures of air and aqueous
vapour is added.
In the section on Combustion, new and simple formulas and data
are given for the quantity of air consumed in combustion, and of
the gaseous products of combustion, the heat evolved by combus
tion, the heating power of combustibles, and the temperature of
combustion ; with several tables.
On Coal as a Fuel, both English and Foreign, its composition,
with the results of many series of experiments on its combustion,
are collected and arranged. Thequantityof air consumed in its com
bustion, and of the gaseous products, with the total heat generated,
are calculated in detail. Coke, lignite, asphalte, wood, charcoal,
peat, and peatcharcoal, are similarly treated ; whilst the combus
tible properties of tan, straw, liquidfuels, and coalgas, are shortly
treated.
The section on Strength of Materials is wholly new. The great
accumulation of experimental data has been explored, and the most
important results have been abstracted and tabulated. The results
of the experiments of Mr. David Kirkaldy occupy the greater por
tion of the space, since he has contributed more, probably, than any
other experimentalist to our knowledge of the Strength of Materials.
The Author has investigated afresh the theory of the transverse
strength and deflection of solid beams, and has deduced a new and
simple series of formulas from these investigations, the truth of which
has been* established with remarkable force by the evidence of experi
ment These investigations, based on the action of diagonal stress,
throw light upon the element called by Mr. W. H. Barlow, "the
resistance of flexure:" revealing, in a simple manner, the nature
of that hitherto occult entity; and showing that flexure is not the
cause, but the effect of the resistance. In addition to formulas
PREFACE. vil
for beams of the ordinary form, special formulas have been
deduced for the transverse strength and deflection of railway
rails, doubleheaded or flanged, of iron or steel; in the estab
lishment of which he has availed himself of the important
experimental data published by Mr. R. Price Williams, and by
Mr. B. Baker. To our knowledge of the strength of timber,
Mr. Thomas Laslett has recently made important additions, and the
results of his experiments have been somewhat fully abstracted and
analyzed. But woods, by their extremely variable nature, are not
amenable, like wroughtiron and steel, to the unconditional applica
tion of formulas for transverse strength. The Author has, never
theless, deduced from the evidence, certain formulas for the trans
verse strength and deflection of woods, with tables of constants,
which, if applied with intelligence and a knowledge of the uncer
tainties, cannot fail to prove of utihty.
The Torsional Strength of Solid Bodies has also been investigated
afresh, and reduced to new formulas.
In dealing with the Strength of Elementary Constructions, the
Author has brought together many important experimental results.
In treating of rivetjoints and their employment in steamboilers,
he has, he believes, clearly developed the elements of their strength
and their weakness. By a close comparison of the results of tests of
castiron flanged beams, it is plainly shown that the ultimate
strength of a castiron beam is scarcely affected by the proportionate
size of the upper flange, and that the lower flange' and the web are,
practically, the only elements which regulate the strength. The
tests of solidrolled and rivetted wroughtiron joists are also ana
lyzed ; and for the strength and deflection of these, as for those of
castiron flanged beams, new and simple rules and formulas are
given, A new investigation, with appropriate formulas, is given
for the bursting strength of hollow cylinders, of whatever thickness.
It is shown that the variation of stress throughout the thickness,
follows a diminishing hyperbolic ratio from the inner surface to
wards the outer surface. The resistance of tubes and cylindrical
flues to collapsing pressure is also investigated, and formulas based
on the results of experience are given.
On the subject of Millgearing, a new and compact table of the
pitch, number of teeth, and diameter of toothed wheels is given,
with new formulas and tables for the strength and horsepower of
the teeth of wheels, and for the weight of toothed wheels. New
formulas and tables are given for the driving power of leather
viii • PREFACE.
belts, and the weight of castiron pulleys. For the strength of
Shafting, — castiron, wroughtiron, and steel, — a new and complete
series of formulas has been constructed, comprising its resistance
to transverse deflection and to torsion, with very full tables of the
weight, strength, power, and span of shafting.
The Evaporative Performance of Steamboilers is exhaustively
investigated with respect to the proportions of fuel, water, grate
area, artd heatingsurface, and the relations of these quantities are .
reduced to simple formulas for different types of boilers. The
actual evaporative performances of boilers are abstracted in tabular
form, comprising those of the Lancashire and the Galloway boilers
at Wigan, tested by Mr. Lavington E. Fletcher.
The Performance of Steam worked expansively, in single and
in compound cylinders, is exhaustively analyzed by the aid of
diagrams ; the similarity and the dissimilarity of its action in the
Woolfengine and the Receiverengine, are investigated; and the
principles of calculation to be applied respectively to these, the
leading classes of compound engines, are explained. The best
working ratios of expansion are deduced from the results of numerous
experiments and observations on the performance of steamengines.
The principles of Aircompressing Machines, and Compressedair
Engines, — a branch of mechanical practice of comparatively recent
origin, — are investigated, and convenient formulas and tables for
use are deduced.
Such are some of the new features of this volume. It may be
added, that the other portions of the Work, likewise, have been
carefully elaborated. The whole of the materials for its preparation
have been drawn from the best available sources, foreign as well as
English. Vast stores of the results of experience are accumulated
in the Proceedings of t/te Institution of Civil Engifieers^ the
Proceedings of tfie Institution of Mechanical Engineers, as well as
in various periodicals and journals English and foreign. From
these and other acknowledged sources, the Author has drawn
much of his material.
D. K. CLARK.
8 Buckingham Street, Addphi,
London, 20th March, 1877.
CONTENTS.
GEOMETRICAL PROBLEMS. .
PACB
Straight Lines — Straight Lines and Circles— Circles and Rectilineal Figures — The
Ellipse — The rarabola — The Hyperbola — The Cycloid and Epicycloid —
The Catenary — Circles — Plane Trigonometry — Mensuration of Surfaces —
Solids — Heights and Distances, I
MATHEMATICAL TABLES.
Explanation of the following Tables : — 32
Logarithms of Numbers from i to 10,000, 38
Hyperbolic Logarithms of Numbers from 1. 01 to 30, 60
Numbers or Diameters of Circles, Circumferences, Areas, Squares, Cubes,
Square Roots, and Cube Roots, .66
Circles : — Diameter, Circumference, Area, and Side of Equal Square, . 87
Lengths of Circular Arcs from 1° to iSo**, 95» 97
Areas of Circular Segments, 100
Sines, Cosines, Tangents, Cotangents, Secants, and Cosecants of Angles, . . 103
Logarithmic Sines, Cosines, Tangents, and Cotangents of Angles, . . 1 10
Rhumbs, or Points of the Compass, 117
Reciprocals of Numbers from I to 1000,  118
WEIGHTS AND MEASURES.
Water as a Standard — Weight and Volume of pure Water — The Gallon and other
Measures of Water — Pressure of Water — Seawater — Ice and Snow — French
and English Measures of Water, 1 24
Air as a Standard — Pressure of the Atmosphere — Measures of Atmospheric Pres
sure — ^Weight of Air — Volume — Specifie Heat, 127
Great Britain and Ireland — Imperial Weights and Measures, .128
Measures of Length : — Lineal — Land — Nautical— Cloth, 129
Wiregauges, 130
Inches and their Equivalent Decimal Values in part^of a Foot — Fractional Parts
of an Inch, and their Decimal Equivalents, 135
Measures of Surface : — Superficial — Builders' Measurement— Land, . .136
Measures of Volume : — Solid or Cubic — Builders* Measurement, . • *37
Table of Decimal Parts of a Square Foot in Square Inches, . . .138
Measures of Capacity : — Li<uid— Dry — Definition of the Standard Bushel — Coal
— Old Wme and Spirit — Old Ale and Beer — Apothecaries* Fluid, . 138
Measures of Weight: — Avoirdupois — Troy — Diamond — Apothecaries* — Old
Apothecaries' — Weights of Current Coins — Coal— Wool — Hay and Straw
— Com and Flour, 140
Miscellaneous Tables : — Drawing Papers — Commercial Numbers — Stationery —
Measures relating to Building — Commercial Measures — Measures for
Ships, 143
Comparison of English Compound Units : — ^Measures of Velocity — Of Volume
and Time — Of Pressure and Weight — Of Weight and Volume — Of Power, 144
X CONTENTS.
PAGE
France — The Metric Standards of Weights and Measures — Metre— Kilogramme, . 146
Countries where the system is legalized, . . . . . • « • .146
Measures of Length, 147
Wiregauges, 148
Measures of Surface, 149
Measures of Volume : — Cubic — Wood, 149
Measures of Capacity : — Liquid — Dry, 149
Measures of Weight, 150
Equivalents of British Imperial and French Metric Weights and Measures, . 150
Measures of Length — Tables of Equivalent Values of Millimetres and Inches —
Square Measures or Measures of Surface — Cubic Measures — Wood Mea
sure — Measures of Capacity — Measures of Weight, 150
Approximate Equivalents of English and French Measures, 156
Equivalents of French and English Compound Units of Measurement : — Weight,
Pressure, and Measure — Volume, Area, and Length — Work — Heat — Speed —
Money, 157
German Empire : Weights and Measures : — Length — Surface — Capacity — Weight, 160
Values of the German Fuss or Foot in the various States, 161
Old Weights and Measures in Prussia (Kingdom of) — Bavaria (Kingdom of) —
Wiirtembeig (Kingdom of) — Saxony (Kingdom of) — Baden (Grandduchy of)
— Hanse ']u)wns: — Hamburg — Bremen — Lubec — Old German Customs
Union — Oldenburg — Hanover, &c., 162
Austrian Empire, 170
Russia, 171
Holland — Belgium — Norway and Denmark — Sweden 173
Switzerland — Spain — Portugal — Italy 175
Turkey — Greece and Ionian Islands — Malta, 178
Egypt — Morocco — Tunis — Arabia — Cape of Good Hope, 179
Indian Empire — Bengal — Madras — Bombay — Ceylon, 180
Burmah — China— CochinChina — Persia — ^Japan — ^Java, 183
United States of North America, 186
British North America, 187
Mexico — Central America and West Indies — West Indies (British) — Cuba — Guate
mala and Honduras — British Honduras — Costa Rica — St. Domingo, . . 187
South America — Colombia — Venezuela — Ecuador — Guiana — Brazil — Peru — Chili —
Bolivia — Argentine Confederation — Uruguay — Paraguay, .... 188
Australasia : — New South Wales — Queensland — Victoria — New Zealand, &c., . 189
MONEY— BRITISH AND FOREIGN.
Great Britain and Ireland : — Value, Material, and Weight of Coins — Mint Price of
Standard Gold, &c., 190
France: — Material and Weight of French Coins, and Value in English Money, . 190
German Empire : — Names and Equivalent Values of Coins, 191
North and South Germany (Old Currency of), 191
Hanse Towns (Old Monetary System of): — Hamburg, Bremen, Lubec, . . . 191
Austria — Russia — Holland — Belgium — Denmark — Sweden — Norway, . . . 192
Switzerland — Spain — Portugal — Italy — Turkey — Greece and Ionian Islands — Malta, 1 93
Egypt — Morocco — Tunis — Arabia — Cape of Good Hope, 194
Indian Empire — China — CochinChina — Persia — ^Japan — Java, .... 195
United States of North America, 195
Canada — British North America, 196
Mexico— West Indies (British) — Cuba — Guatemala — Honduras — Costa Rica —
St. Domingo, 196
CONTENTS. XI
PACK
Sooth America — C olombia — Venezuela — Ecuador — Guiana — BrazU — Peru — Ch ili
— Bolivia — ATgentine Confederation — Uniguay — Paraguay, . . . igt
Austialasia, . .  197
WEIGHT AND SPECIFIC GRAVITY.
Standard Bodies and Temperatures for Comparative Weight — Rules for Specific
Gravity, u 198
General Comparison of the Weights of Bodies, •199
Tables of the Volume, Weight, and Specific Gravity of Metallic Alloys — Metals —
Stones, 200
Coal — Peat — Woods — WoodCharcoal,* 206
Animal Substances — Vegetable Substances, • . 212
Weight and Volume of various Substances, by Tredgold, 213
Weight and Volume of Goods carried over the Bombay, Baroda, and Central
Indian Railway, 213
Weight and Specific Gravity of Liquids, . 215
Weight and Specific Gravity of Gases and Vapours, 216
WEIGHT OF IRON AND OTHER METALS.
Data for Wrought Iron — for Steel— for Cast Iron, . .' . . . .217
Tables of Weights: — Weights of given Volumes of Metals — Volumes of given Weights
of Metals — Weight of One Square Foot of Metals — Weight of Metals of a
given Sectional Area, 218
Special Tables for the Weight of Wrought Iron: —
Rales for the Weight of Wrought Iron — Cast Iron — and Steel, , . . 223
Rule for the Length of I cwt. of Wire of different Metals, of a given thickness, 224
Weight of French Galvanized Iron Wire, 225
Special Tables of the Weight of WroughtIron Bars, Plates, &c. ; Multipliers
for other Metals : — Flat Bar Iron — Square Iron — Round Iron — Angle Iron
and Tee Iron — WroughtIron Plates — Sheet Iron — Black and Galvanized
Iron Sheets — Hoop Iron — Warrington Iron Wire — WroughtIron Tubes,
by Internal Diameter — WroughtIron Tubes, by External Diameter, . 226
Weight of Cast Iron, Steel, Copper, Brass, Tin, Lead, and Zinc — Special Tables : —
CastIron Cylinders, by Internal Diameter — Cast Iron Cylinders, by External
Diameter — Volumes and Weight of CastIron Balls, for given Diameters;
Multipliers for other Metals — Diameter of Cast Iron Balls for given Weights, 253
Weight of FlatBar Steel — Square and Round Steel — Chisel Steel, . . . 259
Weight of One Square Foot of .Sheet Copper — Copper Pipes and Cylinders, by
Internal Diameter — Brass Tubes, by External Diameter — One Square Foot
of Sheet Brass, 261
Siie and Weight of Tin Plates — Weight of Tin Pipes and Lead Pipes — Dimen
sions and Weight of Sheet Zinc, 268
FUNDAMENTAL MECHANICAL PRINCIPLES.
Forces in Equiubrium:— Solid Bodies — Fluid Bodies, 271
Motion : — Uniform Motion — Velocity — Accelerated and Retarded Motion, . . 277
Gravity; — Relations of Height, Velocity, and Time of Fall — Rules and Tables, . 277
Accelerated and Retarded Motion in General: — General Rules— Descent
on Inclined Planes, 282
Mass, 287
Mechanical Centres :— Centre of Gravity — Centre of Gyration— Radius of
Gyration — Moment of Inertia — Centre of Oscillation — The Pendulum —
Length of Seconds Pendulum — Centre of Percussion 287
Central Forces : — Centripetal Force — Centrifugal Force, 294
Xll CONTENTS.
PAGB
Mechanical Elements r—The Lever— The Pulley— The Wheel and Axle—
The Inclined Plane — Identity of the Inclined Plane and the Lever — The
Wedge — The Screw, 296
Work:— English and French Units of Work — Work done by the Mechanical Ele
ments — By Gravity — Work accuxmilated in Moving Bodies — Work done by
Percussive Force, 312
HEAT.
Thermometers: — Table of Equivalent Degrees by Centigrade and Fahrenheit —
Pyrometers, 317, 967
Movements of Heat: — Radiat;ion — Conduction — Convection, .... 329
The Mechanical Theory of Heat: — Mechanical Equivalent of Heat— Joule's
Equivalent in English and French Units — Illustrations, .... 332
Expansion by Heat: — Linear and Cubical Expansion, 335
• Table of Linear Expansion of Solids, 336
Expansion of Liquids,^ . 338
Expansion of Gases — The Absolute Zeropoint — Table of the Compression of
Gases by Pressure under a Constant Temperature, 342
Relations of the Pressure, Volume, and Temperature of Air and other Gases —
General Rules — Special Rules for One round weight of a Gas, with Table
of Coefficients — Table of the Volume, Density, and Pressure of Air at
various Temperatures, . * . . . 346
Specific Heat: — Specific Heat of Water, with Table— Specific Heat of Air
Specific Heat of Solids — Specific Heat of Liquids — Specific Heat of Gases, . 352
Fusibility or Melting Points of Solids: — Table, 363
Latent Heat of Fusion of Solid Bodies, with Rule and Table, .... 367
Boiling Points of Liquids, 368
Latent Heat and Total Heat of Evaporation of Liquids, 370
Boiling Points of Saturated Vapours under various Pressures, . • 37^
Latent Heat and Total Heat of Evaporation of Liquids under One Atmosphere, 372
Liquefaction and Solidification of Gases, 372
Sources of Cold: — Siebe's Ice making Machine — Carre's Cooling Apparatus —
Frigorific Mixtures, 373
STEAM.
Physical Properties of Steam, 378
Gaseous Steam — Its Expansion — Its Total Heat, 383
Specific Heat of Steam — Specific Density of Steam — Density of Gaseous Steam, 384
Properties of Saturated Steam from 32' to 212* F., 386
Properties of Saturated Steam for Pressures of from I lb. to 400 lbs., . 387
Comparative Density and Volume of Air and Saturated Steam, . • 39^^
MIXTURE OF GASES AND VAPOURS.
Respective Pressures of Gas and Vapours in Mixture, 392
Hygrometry, 392
Properties of Saturated Mixtures of Air and Aqueous Vapour, with Table, . 394
COMBUSTION.
Combustible Elements of Fuel — Process of Combustion, 398
Air Consumed in the Combustion of Fuels :— Quantity of the Gaseous Pro
ducts of the Complete Combustion of One Pound of Fuel — Surplus Air, . 400
Heat Evolved by the Combustion of Fuel : — Heat of Combustion of Simple
and Compound Bodies — Heating Powers of Combustibles, .... 402
Temperature of Combustion, 407
CONTENTS. xm
FUELS.
PAGE
Pods or Combustibles generally used, . « 409
Coal: — Its Varieties — Small Coal: — Its Utilization — Washing Small Coal —
Deterioration of Coal by Exposure to Atmosphere, 409
British Coals — Composition of Bituminous Coals — Dr. Richardson's Analysis, . 412
Weight and Composition of British and Foreign Coals, by Delab^che and
Playfair, 413
Weight and Bulk of British Coals, 416
Hygroscopic Water in British Coals, 416
Torbanehill or Boghead Coal, with Table of its Composition, . . . > 417
American and Foreign Coals : — Composition, Weight and Bulk, . . .418
French Coals: — Utilization of the Small Coal — Composition of French Coals —
Mean Density, Composition, and Heating Power, 420
Indian Coals : — Australian and Indian Coals — Composition, ..... 423
Combustion of Coal : — Process of Combustion — Gaseous Products of the Com
bustion of Coal — Surplus Air — Total Heat of Combustion of British Coals, . 426
CoK£ : — Proportion of Coke from Coals — Anthracitic Coke — Weight and Bulk of
Coke — Composition of Coke — Moisture in Coke — Heating Power of Coke, . 430
LiCNiT£ AND Asphalts : — Density, Composition, and Heating Power of Lignites
and Asphaltes, 436
Wood: — Moisture in Wood — Composition — Weight and Bulk of Wood, with
Table — Firewood— Quantity of Air Chemically Consumed in the Complete
Combustion of Wood — Gaseous Products — Total Heat of Combustion —
Temperature of Combustion, . . 439
WoodCharcoal : — ^Yield of Charcoal — Composition, with Table of Composition
at various Temperatures — Carbonization of Wood in Stacks, and Yield of
Charcoal — Manufacture of Brown Charcoal — Distillation of Wood — Oiiarbon
de Paris (artificial fuel) — Weight and Bulk of WoodCharcoal — Absolute
Density of Charcoal — Moisture in Charcoal — Air Consumed in the Combus
tion of Charcoal — Gaseous Products — Heat of Combustion, . . . 444
Peat: — Nature and Composition — Condensed Peat — Average Composition — Pro
ducts of Distillation — Heating Power of Irish Peat, 452
PeatCharcoal: — Composition and Heating Power, 455
Tan : — Composition and Heating Power, , 455
Straw: — Composition, 456
Liquid Fuels : — Petroleum, PetroleumOils, Schist Oil, and Pinewood Oil ; their
Composition and Heating Power, 456
CoalGas : — Composition and Heating Power, . . . ... • • 457
APPLICATIONS OF HEAT.
Transmission of Heat through Solid Bodies — from Water to Water
THROUGH Solid Plates and Beds: — M. Peclet's Experiments— Mr. James
R. Napier's Experiments — Circumstances which affect the Ratio of Trans
misaon — Mr. Craddock's Experiments, 459
Heating and Evaporation of Liquids by Steam through Metallic
Surfaces: — Experiments by Mr. John Graham, by M. Clement, by M.
Pcclet, by MM. Laurens and Thomas, by M. Havrez, by Mr. William
Anderson, by Mr. F. J. Bramwell — Table of Performance of Coiled Pipes
and Boilers in Heating and Evaporating Water by Steam, with Deductions, 461
CoouNG of Hot Water in Pipes: — Observations of M. Darcy — Experiments
by Tredgold — Deductions, 469
Cooling of Hot Wort on Metal Plates in Air: — Results of Experiments
at Trueman*s Brewery, 470
CoouNG OF Hot Wort by Cold Water in Metallic Refrigerators: —
Table of Results of Performance, and Deductions, 471
XIV coNTE^r^s.
PAGE
Condensation of. Steam in Pipes Exposed to AiR:—Expcriments by Tred
gold, and by M. Burnat, on Pipes with various Coverings, with Table —
Experiments by Mr. B. G. Nichol, by M. Clement, by M. Grouvelle —
Condensation of Steam in a Boiler Exposed in Open Air, .... 472
Condensation of Vapours in Pipes or Tubes by Water:— M. Audenet's
Experiments on Steam — Mr. B. G. Nichol's Experiments — Condensation of
other Vapours, 475
Warming AND Ventilation: — Allowance of Air for Ventilation, . . . 477
Ventilation of Mines by Heated Columns of Air.— Furnace Ventilation
— Mr. Mackworth's Data 479
Cooling Action of WindowGlass:— Mr. Hood's Data, .... 480
Heating Rooms by Hot Water: — Mr. Hood's Estimates — Total Quantity of
Air to be Warmed per Minute — Table of the Length of 4inch Pipe required
to Warm any Building — Boilerpower — French Practice — Perkins' System, . 481
Heating Rooms by Steam: — Length of 4inch Pipe required — French Practice, 486
Heating by Ordinary Open Fires and Chimneys:— M. Claudel's Data, . 488
Heating by Hot Air and Stoves: — Sylvester's CockleStove — French Prac
tice — House Stoves placed in the Rooms to be Warmed — House Stoves
placed outside the Rooms to be W^armed, 488
Heating of Water by Steam in Direct Contact: — Mr. D. K. Clark's
Experiments, 490
Evaporation (Spontaneous) in Open Air: — Dalton's Experiments, and Detluc
tions — Rule for Spontaneous Evaporation — Dr. Pole's Formula, . . '491
Desiccation by Dry Warm Air: — Design of a Diying Chamber— Results of
Experiments — Dryinghouse for Calico — Drying Linen and Various Stuffs —
Drying Stuffs by Contact with Heated Metallic Surfaces — Drying Grain —
Drying Wood, 493
Heating of Solids: — Cupola Furnace — Plaster Ovens — Metallurgical Furnaces
— IMast Furnaces, 497
STRENGTH OF MATERIALS.
Definitions 500
Work of Resistance of Material, 501
Coefficient of Elasticity, 503
Transverse Strength of Homogeneous Beams, 503
Symmetrical Solid Beams: — Investigation and Generalized Formula, . . 503
Formula for the Transverse Strength of Solid Beams of Symmetrical Section,
without Overhang, and Flanged or Hollow — For Unsymmetrical Flanged
Beams — Neutral Axis — Elastic Strength, 509
Forms of Beams of Uniform Strength:— SemiBeams Loaded at One End
I — Uniformly Loaded, 517
Forms of Beams of Uniform Strength, Supported at Both Ends — Under a Con
centrated Rolling Load, 521
Shearing Stress in Beams and PlateGirders 525
Deflection of Beams and Girders :— Investigation — Rectangular Beams—
Doubleflanged — Uniform Beams Supported at Three or more Points, . . 527
Torsional Strength of Shafts:— Round— Hollow— Square— Deflection, . 534
Strength of Timber:— Results of Experiments, 537
Transverse Strength of Timber of Large Scantling, 542
Elastic Strength and Deflection of Timber: — Experiments by MM. Chevandier
and Wertheim, by Mr. Laslett, by Mr. Kirkaldy, by Mr. Barlow, . . 545
Rules for the Strength and Deflection of Timber, 548
Strength of Cast Iron: — Tensile Strength and Compressive Strength Results
of Experiments, 553
Shearing Strength, 561
CONTENTS. XV
PAGB
Tmnsvcrse Strength: — Results of Experiments — Test Bars — Transverse Deflection
and Elastic Strength, . . / 561
Torsional Strength, 565
Strekgth of Wrought Iron :~Tensile Strength, &c.— Mr. Kirkaldy's Experi
ments, 567
Experiments of the Steel Committee of Civil Engineers, 579
Hammered Iron Bars (Swedish) — Knipp and Yorkshire Plates— Piussian Plates, 581
Iron Wire, 586
Shearing and Punching Strength, 5S7
Transverse Strength — Deflection and Elastic Strength, 588
Torsional Strength, 590
Strength of Steel:— Mr. Kirkaldy*s Early Experiments— Hematite Steel—
Krupp Steel, 593
Experiments of the Steel Committee, 596
Experiments at H.M. Gun Factory, Woolwich — Fagersta Steel, Mr. Kirkaldy's
Experiments, in seven series, 604
Siemens Steel Plated and Tyres — Mr. Kirkaldy's Experiments, . . .612
Whitworth's Fluid compressed Steel, 614
Sir Joseph Whitworth's Mode of Expressing the Value of Steel, . .615
ChemofTs Experiments on Steel, 616
Steel Wire, 617
Shearing Strength of Steel, 617
Transverse Strength and Deflection, 617
Torsional Strength, 619
Strength Relatively to the Proportion of Constituent Carbon, . . . .621
Resistance of Steel and Iron to Explosive Force > . 622
Recapitulation of Data on the Direct Strength of Iron and Steel: —
Tensile and Compressive Strength of Cast Iron, Wrought Iron, and Steel —
Diagram of the Relative Elongation of Bars of Cast Iron, Wrought Iron, and
Steel, 623
Working Strength of Materials— Factors of Safety: — Factors of Safety
for Cast Iron, Wrought Iron, Steel, and Timber — Load on Foundations,
Masonwork — Ropes — Dead Load — Live Load, 625
Tensile Strength of Copper and other Metals:— Tables of the Strength
of Copper and its Alloys: Tin, Lead, Zinc, Solder, 626
Tensile Strength of Wire of Various Metals:— Tenacity of Metallic
Wires at Various Temperatures — Wires of Various Metals, . . 628
Strength op Stone, Bricks, &c. : — ^Table of the Tensile Strength of Sandstones
and Grits, Marbles, Glass, Mortar, Plaster of Paris, Portland Cement, Roman
Cement, Granites, Whinstone, Limestone, Slates, Bricks, Brickwork in
Cement — Adhesion of Bricks, 629
STRENGTH OF ELEMENTARY CONSTRUCTIONS.
RivetJoints: — In Iron Plates, 633
In Steel Plates, 642
PiLUiRS or Columns : — Compressive Strength 643
CastIron Flanged Beams: — Transverse Strength, 647
Deflection and Elastic Strength, 652
WroughtIron Flanged Beams or Joists: — Solid Wroughtiron Joists —
Transverse Strength and Deflection, 653
Riretted Wroughtiron Joists, 657
Bt'CKLED Iron Plates, 660
I
XVI CONTENTS.
PACB
Railway Rails: — Transverse Strength of Rails of SynMnetrical Section, . .661
Rails of Unsymmetrical Section, . 665
Deflection of Rails, 668
Steel Springs: — ^Laminated and Helical, 671
Ropes: — Hemp and Wire, 673
Chains, 677
Leather Belting, 679
Bolts and Nuts 680
Screwed StayBolts and Flat Surfaces 685
Hollow Cylinders — Tubes, Pipes, Boilers, &c.: — Resistance to Internal or
Bursting Pressure — Transverse Resistance, 687
Longitudinal Resistance to Bursting Pressure, 692
Wroughtiron Tubes, 693
Castiron Pipe 693
Resistance to External or Collapsing Pressure — Solid drawn Tubes — Large Flue
Tubes — Lead Pipes, 694
Framed Work — Cranes, Girders, Roofs, &c.: — The Triangle the Funda
mental Feature, 697
WarrenGirder Loaded at the Middle, and at an Intermediate Point — Uniformly
Loaded — Rolling Load, 699
Parallel 1atticeGirder. 708
Parallel StrutGirder, 708
Roofs, 713
WORK, OR LABOUR.
Units of Work or Labour: — Horsepower — Mechanical Equivalent of Heat
— Labour of Men, 718
Labour of Horses — Work of Animals Carrying Loads, 720
FRICTION OF SOLID BODIES.
Laws of Friction:— Friction of Journals — Friction of Flat Surfaces, . . 722
Friction on Rails: — M. Poiree's Experiments 724
Work and Horsepower Absorbed by Friction:— Formulas, . . . 725
MILLGEARING.
Toothed Gear:— Pitch of the Teeth of Wheels— Spur Flywheels— Toothed
Wheels for Millwork — Rules, 727
Form of the Teeth of Wheels, . . .731
Proportions of the Teeth of Wheels, 734
Transverse Strength of the Teeth of Wheels — Working Strength, . . . 735
Breadth of the Teeth of Wheels, 737
Horsepower Transmitted by Toothed Wheels, 737
Weight of Toothed Wheels, . . 739
Frictional WheelGearing, 741
Belt Pulleys and Belts.— Tensile Strength, 742
Horsepower Transmitted by Belts, 743
Adhesion and Power of Belts — Examples of very wide Belts, .... 744
Indiarubber Belting, 750
Weight of Belt Pulleys, 750
Rope Gearing: — Transmission of Power by Ropes to Great Distances, . . 753
Cotton Ropes^ . , , . , 755
CONTENTS. XVll
PAGB
Shafting: — ^Transverse Deflection of Shafts, ,756
Ultimate Torsional Strength of Round Shafts, 758
Torsional Deflection of Round Shafts, 759
Power Transmitted by Shafting 760
Weight of Shafting, 761
Strength and Horsepower of Round Wroughtiron Shafting, .... 762
Frictional Resistance of Shafting, 763
Ordinary Data for the Resistance of Shafting, ... ... 763
Joomals of Shafts, 766
EVAPORATIVE PERFORMANCE OF STEAMBOILERS.
Normal Standards, 768
Heating Power of Fuels:— Table of Heating Power, 769
E%'APORATivE Performance of Stationary and Marine Steam  Boilers,
WITH Coal: — Surplus Air Admitted to the Furnace, .... 770
Experiments on the Evaporative Power of British Coals, by Delab^che and
Playfair, 770
Evaporative Performance of Lancashire Stationary Boilers at Wigan — With
Economizer and Without Economizer ~  Watertubes — Temperature of
the Products of Combustion, and of the Feedwater — Trials of D. K.
Clark's SteamInduction Apparatus — Of Vicars' Self feeding Firegrate, . 771
Evaporative Performance of South Lancashire and Cheshire Coals in a Marine
Boiler, at Wigan 781
Trials of Newcastle and Welsh Coals in the Wigan Marine Boiler, . . 784
Evaporative Performance of Newcastle Coals in a Marine Boiler, at Newcastle
onTyne, 785
Trials of Newcastle and Welsh Coals in the Marine Boiler at Newcastle, for the
Board of Admiralty, 787
Trials of Welsh and Newcastle Coals in a Marine Boiler at Keyham Factory, . 790
Evaporative Performance of American Coals in a Stationary Boiler, . . . 791
Evaporative Performance of an Experimental Marine Boiler, Navy Yard, New
York, 795
Evaporative Performance of Stationary Boilers in France, .... 796
Evaporative Performance of Locomotive Boilers, 798
Evaporative Performance of Portable Engine Boilers, 801
Relations of GrateArea and Heating Surface to Evaporative Per
formance: — Mr. Graham's Experiments — Experiments by Messrs. Woods
and Dewrance — Experimental Deductions of M. Paul Havrez, . . . 802
Formulas for the Relations of GrateArea, Heating Surface, Water,
and Fuel:— General Equations, 804
Formulas for the Experimental Boilers, 807
General Formulas for Practical Use, 819
Table of the Equivalent Weights of Best Coal and Inferior Fuels, . . . 820
STEAMENGINE.
Action of Steam in a Single Cylinder:— The Work of Steam by Expan
sion — Clearance — Formulas for the Work of Steam — Initial Pressure in the
Cylinder — Average Total Pressure in the Cylinder — Average Effective Pres
snre — Period of Admission and the Actual Ratio of Expansion — Relative
Performance of Equal Weights of Steam Worked Expansively — Proportional
Work Done by Admission and by Expansion — Influence of Clearance in
Redndng the Performance of Steam 822
Table of Ratios of Expansion of Steam, with Relative Periods of Admission,
Pressures, and Total Performance, 835
b
XVlll CONTENTS.
PACK
Total Work Done by One Pound of Steam Expanded in a Cylinder, . 838
Consumption of Steam Worked Expansively per Horsepower of Net Work
per Hour, 840
Table of the Work Done by One Pound of Steam of lOOlbs. Pressure per
Square Inch, 841
Net CylinderCapacity Relative, to the Steam Expanded and Work Done in
One Stroke, 843
Table of ditto, 844
Compound SteamEngine; — Woolf Engine— ReceiverEngine— Ideal Diagrams,
without Clearance — Work of Steam as Affected by Intermediate Expansion
— Intermediate Expansion — Work, with Clearance — Comparative Work of
Steam in the Wool! Engine and the Receiver Engine, 849
Formulas and Rules for Calculating the Expansion and the Work of Steam, . 869
Compression of Steam in the Cylinder, 878
Practice of the Expansive Working of Steam: — Actual Performance —
Data — Deductions — Conclusions, 879
FLOW OF AIR AND OTHER GASES.
Discharge of Air through Orifices — Anemometer, 891
Outflow of Steam through an Orifice, 893
Flow of Air through Pipes and Other Conduits 894
Resistance of Air to the Motion of FlatSurfaces 897
Ascension of Air by Difference of Temperature, 897
WORK OF DRY AIR OR OTHER GAS, COMPRESSED OR
EXPANDED.
Work at Constant Temperatures:— Isothermal Compression or Expansion, 899
Work in a Nonconducting Cylinder, Adiabatically, .... 901
Efficiency of CompressedAir Engines, 909
Compression and Expansion of Moist Air, 912
AIR MACHINERY.
Machinery for Compressing Air and for Working by Compressed
Air: — Compression of Air by Water at Mont Cenis Tunnel Works — By
Directaction Steampumps — Compressedair Machinery at Powell Duffryn
Collieries, 915
Hot Air Engines: — Laubereau's— Rider's — Belou*s — ^Wenham's, . . . 917
GasEngines: — Lenoir^s — Hugon*s — Otto & Langen's, 920
Fans or Ventilators: — Common Centrifugal Fan— Mine Ventilators — GuibaFs
Fan — Cook's Ventilator, 924
Blowing Engines 926
Root's Rotary PressureBlowers, 92^
FLOW OF WATER.
Flow of Water through Orifices: — Formulas — Mr. Bateman's Experi
ments, 929
Mr. Brownlee's Experiments with a Submerged Nozzle, 931
Flow of Water Ovi;R WasteBoards, Weirs, &c., 932
Flow of Water in Channels, Pipes, and Rivers, 932
CastIron Water Pipes, 934
C ASTIron Gas Pipes, 936
CONTENTS. • XIX
WATERWHEELS.
PAGE
Wheels on a Horizontal Axis:— Undershot Wheels— Paddle Wheels— Breast
Whcels — Overshot Wheels, 937
Wheels on a Vertical Axis:— Tub— Whitelaw*s Watermill— Turbines
Tangential Wheels 939
MACHINES FOR RAISING WATER.
Pumps: — Reciprocating Pumps — Centrifugal Pumps — Chain Pump — Noria, . 944,. 968
Water works Pumping Engines, . . . . 948
Hydzanlic Rams, 949
HYDRAULIC MOTORS.
Hydraulic Press, 950
Armstrong's Hydraulic Machines, 950
FRICTIONAL RESISTANCES.
Steam Engines, 951
Tools: — Shearing Machines — Platebending Machines — Circular Saws, '951
VTork of Ordinary Cutting Tools, in Metal, 952
Screwcutting Machines — Woodcutting Machines — Grindstones, . 954
Colliery Winding Engines, 956
Waggons in Coal Pits, 956
Machinery of Flax Mills: — ^M. Comut*s Experiments, 957
Hoisex>wer Required, 959
Machinery of Woollen Mills:— Dr. Hartig's Experiments, . . . , . 959
Machinery for the Conveyance of Grain, 960
Traction on Common Roads: — M. Dupuit's Experiments — M. Debauve's De
ductions — M. Tresca's Experiments, 961
Carts and Waggons on Roads and on Fields, 962
Resistance on Railways, 965
Resistance on Street Tramways, 966
APPENDIX.
Dr. Siemens' Water Pyrometer, 967
Atmospheric Hammers, 967
Bernays* Centrifugal Pumps, 968
SteamVacuum Pump, . 969
Index, .  .'.... 971
AUTHORITIES CONSULTED OR QUOTED.
American, United, Railway Master Car
Builders* Association, Standard Sizes of
Bolts and Nuts by, 663.
American Society of Civil Engineers, Journal
of: — Mr. J. F. Flagg, on Steamvacuum
Pamps, 969.
Anderson. Dr., on the Strength of Cast Iron,
555
Anderson, William, on Heating Water by
Steam, 465. 466, 468 ; Translation of Cher
noffs Paper on Steel, 616.
Annales dts Mines: — M. Krest, on the Slip
of Belts, 742.
Annales des Fonts et Chaussies: — M. Him's
Rope Transmitter of Power, 754.
Annales du Ginie Civile: — M. Paul Havrez,
on Heating Surface of Locomotives, 803.
A mnals of Philosophy : — Mr. Dunlop, on Tor
sional Strength of Cast Iron, 565.
Annuaire det Association des Inginieurs sort is
de r^coU de Zi^/.— Rivelted Joints, 641.
Armengaud, French Standard Bolts and
Nuls, by, 683.
Aimstrong. Sir Wm., on Evaporative Power
of Coals, 785; his Hydraulic Machinery,
950
Arson, Anemometer by. 892.
A^by & Co., Work of Steam in Portable
Engine by, 883.
Audenet, on SurfaceCondensers, 475.
B
BaJcer. B., on the Strength of Beams, 512;
of Oak. 544, 549; of Columns, 645, 646;
of RaQs, 662, 666.
Barlow. Peter, on Strength of Timber, 547 ;
of Cast Iron, 561 ; of Wrought Iron, 567,
588, 590 ; of Iron Wire, 586.
Bariow. W. H,. on the " Resistance of Flex
ure." 507.
Bamaby. Mr., on Strength of Punched Steel
Plate, 642.
Barrow Hematite Steel Company, Strength
of Steel made by, 594. 618, 619, 620, 621.
Batcman, J. F., on Flow of Water through
Submerged Openings. 930; his CastIron
Pipes. 934
Baudrimont, on Strength of Metallic Wires.
628.
Beardmore, on the Work of Horses, 720 ; on
Limits of Velocity at the Bottom of a
Channel, 934.
Beaufoy, Colonel, on Resistance of Air, 897.
Bell, J. Lothian, on the Heat in Blast Fur
naces, 498.
Berkley, George, on the Strength of Cast
iron Beams. 647650.
Berkley, J., Specific Gravity of Indian Woods,
by, 209.
Bemays, Joseph, on Centrifugal Pumps, 968.
Bertram, W., on Rivetted Joints, 634637.
Borsig, Herr, Strength of Wrought Iron
Plates, 586.
Box, Thomas, on the Load on Journals, 766 ;
Thickness of Gas Pipes, by, 936.
Boyden, Outflow Turbine by, 940.
Bradford, W. A., on Otto and Langen's Gas
Engine. 924.
Bramwell. F. J., on Heating Water by Steam,
467, 468: on the Strength of Cast Iron,
556 ; on Portable Steam Engines, 801, 883,
886 ; on the Expansive Working of Steam,
889.
Brereton, R. P., on Strength of Timber Piles,
646.
Briggs, Blowing Engine by, 927.
British Associatiom, Transactions of : — F. W.
Shields, on Strength of Castiron Columns.
645
Brown & May, Work of Steam in Portable
Engine by, 882.
Brownlee, J., on Saturated Steam, 382; on
the Outflow of Steam, 893 ; Flow of Water
through a Submerged Nozzle, 931.
Bruce, G. B., on the Work of a Labourer,
719.
Brunei, on the Strength of Rivetted Joints,
638 ; and of Bolts and Nuts, 680.
Buchanan, W. M., on Saturated Steam, 379.
Buckle. W., on Fans, 924.
Buel, R. H., on the Slip of Belts, 742.
Bulletin de la Sociiti Industrielle de Mul
house: — M. Leloutre on Steam Engines,
886.
Burnat, on Condensation of Steam in Pipes,
472. 474
Bury, Wm., on Strength of Flat Stayed Sur
faces, 686.
<52
XXll
AUTHORITIES CONSULTED OR QUOTED.
Cameron, Dr. , Analysis of Peat by, 454.
Chari^Marsaines, on Flemish Horses, 964.
Chenot Ain^, Atmospheric Hammer by, 967.
Chemoff, on Steel, 616.
Chevandier, on Composition of Wood, 440;
on its Weight and Bulk, 442, 443.
Chevandier & Wenheim, on Strength of Tim
• ber, 538, 545, 546, 549.
Clark, D. t^., on Proi>erties of Saturated
Steam, 387; on Locomotive Boilers, 798;
on the Work of Steam. 879, 880^ 884; on
Resistance on Railways, 965.
Clark, Edwin, on the Strength of Beams, 51a;
of Red Pine, 543, 544, 549 ; of Cast Iron,
562 ; of Bar Iron, 570, 588, 590, 623.
Clark, Latimer, on Wire Gauges, 130.
Claudel on Fuels and Woods, by, 207, 211,
212 } tints of Heated Iron, 328 ; on Heating
Factories, 486; on Heating Rooms, 488,
489; on Belts, 743, 746; on Blowing En
gines, 927; on Pumps, 944.
Clement, on Transmission of Heat, 462, 468 ;
on Condensation of Steam in Pipes, 474 ;
on Drying Stuffs, 496 ; on the Heat to Melt
Iron, 497.
Cochrane, J., on Strength of Perforated Bar
Iron, 633.
Cockerill, John, Blowing Engines by, 927.
Colliery Guardian : — Mr. Mackworth on Ven
tilation of Mines. 480.
Conservatoire cUs Arts et Metiers, Annates
du: — HotAir Engines by Laubereau. and
by Belou, 9179x9 ; GasEngines by Lenoir,
920; by Hugon, 921; by Otto & Langen,
923
Cooper, J. H., on Very Wide Belts, 747, 749.
Comet, on the Work of a Labourer in France,
720.
Comut, E., on MillShafting, 766 ; on Machin
ery of FlaxMills, 957 ; on Flow of Air in
Pipes, 896.
Cotterill. J. H., on Work of Compression of
Air, 903.
Cowper, E. A., Compound Engine by, 889.
Craddock, Thomas, on Cooling through
Plates, 461.
Crighton & Co. , on Drying Grain, 496.
Crookewitt, on Specific Gravities of Alloys,
2CO.
Crossley, F. W., on Otto & Langen's Gas
Engines, 923.
Cubitt, Mr., on Strength of CastIron Beams,
649.
D
DagUsh, G. H., on Resistance of Colliery
Winding Engines, 956.
Dalton, Dr., on " Spontaneous " Evaporation
of Water, 491.
Daniel, W., on Ventilation of Mines, 925.
Dan vers, F. C, on Coal Economy, 4x0.
Darcy, on Cooling Hot Water in Pipes, 469.
D'Aubuisson, on Flow of Compressed Air,
896 ; on Hydraulic Rams, 949.
Davey, Paxman,. & Co., Work of Steam in
Portable Engine by, 883.
Davies. Thomas, on Strength of Rivetted
Joists. 658.
Davison, R., on Resistance of Shafting,
766; Duty of Pumps by, 944; on Resist
ance of Grain Machinery, 961.
Day, Summers, & Co., Work of Steam in
Marine Engines by, 882.
Debauve, on Resistance on Common Roads,
961.
Delabtehe & Playfair, on British and Foreign
Coals, 206, 413, 416, 770.
Despretz, on Conducting Powers of Bodies,
331
Deville, SainteClaire, on Composition of
Petroleum and other Oils, 456, 457.
Dewrance, John, on the Heating Surface of a
Locomotive, 803.
Donkin, Bryan, & Co., Work of Steam in
Stationary Engines by, 882.
Downing, on Flow of Water in Pipes, 933.
934
Dunlop, on Strength of Cast Iron, 565.
Dupuit, on Resistance on Common Roads,
961.
Durie, James, on RopeGearing, 753.
Duvoir, Ren^, Drying House by, 495.
Eastons & Anderson, on Portable Steam
Engines, "Box ; on Rider's HotAir Engine,
9x7; on Resistance 'of Waggons. 962.
Elder, John, & Co., on the Strength of Boilers.
638, 693 ; Work of Steam in Marine Engine
by, 882.
Emery, on American Marine Engines, 884.
Engineer, ZA*?.— Crighton & Co. on Drying
Grain, 496 ; Mr. W. S. Hall on the Strength
of Rivetted Joints, 641 ; Messrs. Woods &
Dewrance on Locomotive Boilers, 803 ; Mr.
C. L. Hett on Hydraulic Rams, 949.
Engineering: — on Heating Water by Steam,
464; on Cooling Wort, 470, 471; Mr. B.
G. Nichol on Surface Condensation, 476;
Mr. G. Graham Smith on Strength of
Timber, 544; Factor of Safety for Wrought
Iron, by Roebling, 625 ; Mr. W. S. Hall
on the Strength of Rivetted Joints, 64X ;
Mr. John Mason on Strength of Untanned
Leather Belts, 680; Mr. Phillips on Strength
of Flat Plates, 686; Mr. Bury on the Strength
of Flat Stayed Surfaces, 686 ; Messrs. John
Elder & Co. on the Strength of Boilers,
638, 693 ; Mr. J. Durie on Rope Gearing,
^
AUTHORITIES CONSULTED OR QUOTED.
XXlli
753; Dr. Martig on Resistance of Tools,
951; Resistance of Waggons, by Messrs.
Eastons & Anderson, 962.
Emglisk Mechanic : — Mr. W. A. Bradford on
Olto & Langen's GasEngine. 924.
E^Tard« A. on the Work of Animals, 720.
Fagersta Steel Works, Strength of Steel made
at, 604, 618, 619. 690, 621.
Faiifaaim, Sir Williani, on HotBlast Iron,
556; on the Strength of Cast Iron, 557; on
the Strength of Wrought Iron, 567569;
of Rivetted Joints, 633 ; of Screwed Stay
Bolts and Flat Stayed Plates. 685 ; on the
Proportions of Spur Wheels, 729, 734, 737 ;
on the Load on Journals. 766, 767; on
Water Wheels, 938.
Fairbaim & Tate, on the Expansion of Steam,
383
Fairweather, James C, on Resistance of Air,
897
Faraday, Dr., on the Liquefaction of Gases,
372.
Favre & Silbermann, on the Heating Powers
of Combustibles, 404.
Field. Joshua, on the Work of Labourers, 719.
Fincham, on Strength of Timber, 542, 543,
549
Flagg, J. F., on Steamvacuum Pumps, 969.
Fletcher, L. E., on the Strength of a Boiler,
638. 693 ; his Reports. 696 ; his Report on
Bofler and Smoke Prevention Trials, 771
784.
Yowke, Captain, on Colonial Woods. 209.
Fowler, G., on Resistance of Waggons in
Coal Hts, 956.
Fowler, John, Strength of Steel Rails de
signed by, 666, 670.
Fowler, J., & Co., Compressedair Machinery
by, 916.
Fox. Head, & Co., on Condensation of Steam
in a Boiler, 475.
Francis, J. B., on a Swain Turbine, 943.
Franilin Institute, Journal of: — the Shear
ing Resistance of Bar Iron, by Chief
Engineer W. H. Shock, 588; Mr. R. H.
Bud on Belts, 742 ; Mr. H. R. Towne on
Belts, 742, 745; Mr. J. H. Cooper on
Bells, 747: Mr. S. Webber on Mill Shaft
ing. 763, 764; Mr. Emery on American
Marine Engines, 884; Mr. Briggs on
Blowing Engines, 927; Mr. J. B. Francis
on a Swain Turbine, 943; Mr. E. D.
Leavitt's Pumping Engines. 948.
Gammelbo & Co., Hammered Bars made by,
Strength of, 581.
GaudiUot; on Heating Apparatus, 486.
GayLussac. on Cold by Evaporation, 376.
Glynn, Mr. on Strength of Ropes, 673 ; on
the Work of a Labourer, 718.
Gooch, Sir Daniel, on Consumption of Water
by the "Great Britain" Locomotive. 884.
Gordon. L. D. B., on Strength of Colimms,
645
Graham, John, on Heating Water, 461 ; on
Heating Surface, 802.
Grant, on Strength of Cements, &c., 630.
Greaves, on Pumping Engines, 948.
Grouvelle, on Condensation of Steam in Pipes,
474; on Heating Factories, 486. 487.
H
Hackney, W., on Anthracitic Cbke, 432.
Haines, R.. on Indian Coals, 423.
Hall, W. S., on the Strength of Rivetted
Joints, 64X.
Harcourt, Vernon, on Analysis of CoalGas,
458.
Harmegnies, Dumont, & Co., on French Wire
Ropes, 677.
Hartig, Dr., on Driving Belts, 743; on Re
sistance ' of Tools, 951 ; on Resistance of
Machinery of Woollen Mills, 959.
Havrez, P.. on Heating Water by Steam,
464, 468; on Heating Surface of Loco
motives, 803.
Hawksley, Thomas, on Flow of Air through
Pipes, 894 ; on Velocity of Air in Upcast
Shaft, 897 ; on Flow of Water in Pipes.
933 ; on Thickness of Water Pipes, 935.
Hett, C. L., on Hydraulic Rams, 949.
Hick, John, M.P., on Friction of Leather
ColUurs, 950.
Him, on Work of Expanded Steam in Sta
tionary Engines, 886.
Hodgkinson, on the Strength of Cast Iron,
553555. 558, 559. 563. 564; of Columns,
643, 646; of Castiron Flanged Beams,
647650.
Holtzapffel, his WireGauges, 131, 13a, 134.
Hood, on Warming and Ventilation, 477485.
Hopkinson, on the Performance of a Corliss
Engine, 88 z.
Hunt, R., on Combustion of Coal, 770.
Hutton, Dr., Law of Resistance of Air by«
897.
JLa
Institute of Naval Architects, Transactions
of M^.— Strength of Rivet Joints of Sted
Plates, 642.
Institution of Civil Engineers, Proceedings
of:— Mr. Wm. Anderson on Heating Water
by Steam, 465; M. Bumat on Condensation
of Steam in Pipes. 472; Dr. Pole on Spon
taneous Evaporation, 493; Regenerative
HotBlast Stoves. 556: Mr. Bramwell on
XXIV
AUTHORITIES CONSULTED OR QUOTED.
Strength of Cast Iron, 556; Mr. Grant on 
the Strength of Cements, &c., 630; Mr. J.
Cochrane on the Strength of Punched Bar
Iron, 633; Mr. R. Price Williams on
Strength of Rails, 662; Mr. J. T. Smith
on the Strength of Bessemer Steel Rails,
664; Mr. R. Davison on Resistance of
Shafting, 766 ; Evaporative Performance of
Steam Boilers in France, 796 ; Composition
of Coals and Lignites, 797; M. Paul Havrez
on Heating Surface of Locomotives, 803;
Mr. Emery on American Marine Engines,
884; Mr. Hawksley on Flow of Air through
Pipes, 894 ; and on Velocity of Air in Up
cast Shaft, 897 ; M. Piccard on the Work
of Compressed Air, 911 ; Mr. J. B. Francis'
trial of a Swain Turbine, 943; Mr. R.
Davison on Duty of Pumps, 944 ; Hon. R.
C. Parsons on Centrifugal Pumps, 947 ; Mr.
Henry Robinson on Armstrong's Hydraulic
Machines, 950.
Institution of Engineers and ShipBuiiders in
Scotland, Transactions of the: — on Strength
of Helical Springs. 672 ; Report on Safety
Valves, 893; Mr. J. Brownlee's Experi
ments on Flow of Water, 931.
Institution of Mechanical Engineers, Pro
ceedings of: — Mr. C. Little on the Shearing
and Punching Strength of Wrought Iron,
587 ; Mr. Vickers on the Strength of Steel,
621; Mr. W. R. Browne's paper on Rivetted
Joints, 637; Mr. Robertson on Grooved
Frictional Gearing, 741 ; Mr. H. M. Mor
rison on Him's Rope Transmitter, 755;
Mr. Ramsbottom on Cotton Rope Trans
mitter, 755; Mr. Westmacolt and Mr. B.
Walker on Resistance of Shafting, 766 ; Mr.
D. K. Clark on the Expansive Working of
Steam in Locomotives, 879. 880; Data of
the Practical Performance of Steam, 880;
Mr. F. J. Bramwell on Economy of Fuel
in Steam Navigation, 889 ; CompressedAir
Machinery by Messrs. John Fowler & Co.,
916 ; Wenham's HotAir Engine, 919 ; Mr,
F. W. Crossley on Otto and Langen's Gas
Engine, 923; Mr. Buckle on Fans, 924;
Mr.J.S.E. Swindell on Ventilation of Mines,
925; Mr. W. Danielon Ventilation of Mines,
925 ; Mr. A. C. Hill on Blowing Engines,
927 ; Mr. J. F. Bateman's Experiments on
Flow of Water, 930 ; Mr. David Thomson
on Pumping Engines, 948; Mr. G. H.
Daglish on Winding Engines. 956 ; Mr. G.
Fowler on Resistance of Waggons in Coal
Pits, 956 ; Mr. Westmacott on Com Ware
housing Machinery, 961.
Iron and Steel Institute, Journal of the: —
Mr. J. Lothian Bell on the Cleveland Blast
Furnaces, 498.
Isherwood, Trials of Evaporative Performance
of a Marine Boiler, 795.
J
James, Captain, on the Strength of Cast Iron,
555
Jardine, Mr., on the Strength of Lead Pipes,
696.
Johnson, Professor W. R., on American Coals,
418, 770. 791795
Joule, Dr., Mechanical Equivalent of Heat,
by, 332.
K
Kane, Sir Robert, on Peat, 453.
Kennedy, Colonel J. P., on Weight and
Volume of Goods carried on Railways, 213.
Kirkaldy, David, on Compressive Strength
of Timber, 546, 547, 647 ; on the Tensile
Strength of Wrought Iron and Steel, 571
578 ; of Swedish Hammered Bars. 581, 590;
of Krupp and of Yorkshire Iron Plates,
583586 ; of Borsig's Iron Plates, 586; Ten
sile Strength of Bar Steel 593, 594 ; of He
matite Steel, 594 ; of Krupp Steel, 595 ; of
Steel Bars, for the Steel Committee, 597
600; of Fagersta Steel, 604611 ; of Siemens
Steel Plates and Tyres, 612614 ; on Shear
ing Strength of Steel. 617; on Strength of
PhosphorBronre, 628, 629; of Wires, 629;
of Rolled Wroughtiron Joists, 654; of Rails.
662, 663, 666668; of Ropes, 674; of Belt
ing, 680; of Plates of a Marine Boiler, 694.
Krest, on the Slip of Belts, 742.
Krupp, Herr, Strength of WroughtIron Plates
made by, 583 ; of his Cast Steel, 595. 618
62T.
L
Landore SiemensSteel Company, Strength of
Steel Plates and Tyres made by, 6i*6i4.
Laslett. Thomas, on the Strength of Timber,
538542. 546, 548, 550. 647.
Leavitt, E. D., Pumping Engines by. 948.
Legrand, on Boiling Points, 370.
I^igh, Evan, on Belting, 746.
Ldoutre, on M. Him's Experiments on Work
of Steam, 886.
Leplay. on Moisture in Wood, 439; on
Drying Wood, 496.
Literary and Philosophical Society of Man
chester, Memoirs of: — Dr. Dalton on
" Sipontaneous " Evaporation, 491; Mr.
John Graham on Heating Surface, 802.
Little, C, on the Shearing and Punching
Strength of Wrought Iron, 587.
Lloyd, Thomas, on the Strength of Bar Iron.
569. 570.
London Association of Foremen Engineers,
Proceedings of: — Mr. David Thomson on
Expansive Work of Steam, 822.
Longridge, J. A., on Combustion and Evapn
orative Power of Coals, 770, 785.
Longsdon, Mr., on Strength of Krupp Steel,
595
AUTHORITIES CONSULTED OR QUOTED.
XXV
M
MacCon, on the Strength of Rivetted Joints,
641.
Mackintosh. Charles, Weight of BeltPulleys
by. 75a.
Mackworth. H., on Ventilation of Mines,
479
Madure. H. H., on Strength of Timber, 542,
543. 549
Macneil, Sir John, on Resistance on Common
Roads; 964.
Mahan, Lieutenant F. A., on OutwardFlow
Tuibines, 941.
Mallaid, on CompressedAir Machines, 90a ;
on Compressed Air, 907, 91a.
Mallet, R.. Strength of Buckled Iron Plates
by, 660.
Marshall Sons, & Co., Work of Steam in
Portable Engine by, 883.
Mason, John, Strength of Untanned I.^eather
Belts by, 680.
M'Donndl, A, on Composition of Peat, 454.
Menelaus, on Portable Steam Engines, 801.
Miller, T. W., Trials of Coals by, 790.
Miner & Taplin. Trials of Coals by, 787.
Montgoilfier, on Drying by Forced Currents,
494
Monthly Reports to the Manchester Steam
Users' Association: — Mr. L. E. Fletcher's
Data. 696.
Morin. on Transverse Strength of Timber,
537; on the Friction of Journals, 722 ; and 1
of Solid Bodies, 733 ; on Leather Belts,
743745; *5^ Breast Wheels, 938; on a
Foomeyron Turbine. 940; on Centrifugal
Pomps, 946.
Morrison. H. M., on M. Him's Rope Trans
mitter, 755
Morton, Francis, & Co., Weight of Jron
fleets by, 245 ; Strength of Cable Fencing
Stands by, 676.
Moser. Strength of Beams tested for, 654.
Muspratt. Dr.. Analyses of Coke by, 433.
N
Naper, James R., on Transmission of Heat,
460 ; on Drying Stuffs, 496.
Nao. on Moisture in Charcoal, 451.
Xenall. R. S., & Co., Strength of Hemp and
Wire Ropes by, 674.
Nichol, B. G., on Condensation of Steam in
Pipes and Tubes, 474, 476.
Xjcon & Lynn. Trials of Coals by, 784.
Norris & Co.. Strength of Leather Belts by,
68a
North British Rubber Company, Driving Belts
by. 730
Af'orth of England Mining Institute, Transac
tions e^.— Rivetted Joints, 588.
O
Oldham, Dr., on Indian Coals, 424.
Ott, Karl Von, on Strength of Ropes, 674,
679.
P
Parsons, on Strength of Oak Trenails, 551.
Parsons, Hon. R. C, on Centrifugal Pumps,
947
Payen. on Explosive Mixture of Gas and Air,
921.
Pearce, W. A., on Rope Gearing, 754.
Peclet, on Radiation of Heat, 329 ; on French
Coals. 420 ; on Coke, 431 ; on Moisture in
Tan, 455; on Transmission of Heat, 459,
46a, 463, 468; on Condensing Power of
Air and Water, 475 ; on Ventilation, 477 ;
on Heating Apparatus, 488, 489 ; on Drying
by Air Currents, 494 ; on a Drying House,
495 ; on Cupola Furnaces, 497.
Penot, on Drying Houses, 496.
Penrose & Richards, their Anthracitic Coke,
432
Perkins, Heating Apparatus by, 486.
Perkins, Jacob, Invention of the IceMaking
Machine by, 373.
Person, on the Latent Heat of Fusion, 367.
Phillips, on Strength of Flat Plates, 686.
Piccard, on Work of Compressed Air, 911.
Poir^ on Friction on Rails by, 724.
Pole, Dr., on Spontaneous Evaporation, 493 ;
on the Strength of Steel Wire, 617.
Poncelet, on Water Wheels, 938.
Forte/euille de John Cockerill: — Blowing
Engines, 927.
Porter, C. T., on Expansion of Steam, 886.
Pouillet, on Luminosity at High Temper
atures, 328.
R
Radford, R Heber, Weight of BeltPulleys
by,. 751. 752.
Ramsbottom, J., on Cotton Rope Transmitter,
755
Rankine, Dr. , on Expansion of Water, 340 ;
on the Melting Point of Ice. 364; on
Transmission of Heat, 461 ; on Shearing
Strength of Oak Trenails, 551 ; and of
Cast Iron, 561 ; Factors of Safety, 625, 626 ;
on Stresses in Roofs, 715, 717 ; on Load on
Working Surfaces. 767.
Reading Engine Works Co., Work of Steam
in Portable Engine by, 883.
R^clus, Specific Gravity of Sea Water by, 126.
Regnault, Air Thermometer by, 325 ; on the
Expansion of Air, 344 ; on Specific Heat
of Metals, 353; and Gases, 359; Boiling
Points of Vapours, 371 ; on Steam, 378,
379, 383, 384 ; on the Mixture of Gases and
XXVI
AUTHORITIES CONSULTED OR QUOTED.
Vapours, 392 ; on French Coals, 420, 421 ;
on Lignite and Asphalte, 436.
Reilly, Calcott, on the Varieties of Stress,
500
Rennie, on the Work of Horses, 720:
Reime IndustrielU: — Atmospheric Hammer
by M. Chenot Ain^, 967.
Reynolds, Dr., on Peat, 454.
Richardson, Dr., on Coals, 412 ; on Coke,
433; Report on Evaporative Power of Coals,
785
Robertson, James, on Grooved Frictional
Gearing, 741.
Robinson, Henry, on Armstrong's Hydraulic
Machines, 950.
Roebling, on the Strength of Iron Wire, 587 ;
and of Steel Wire, 617 ; Factor of Safety for
Iron, 625 ; on the Strength of Wire Rope
and Hemp Rope, 676.
Ross, Owen C. D., on Coal Gas, 457.
Rouget de Lisle, on Drying StufTs, 496.
Royal Society of Edinburgh, Proceedings of: —
Mr. Fairweather on Resistance of Air, 897.
Royer, on Drying Houses, 496; on Drying
Stuffs, 496.
Russell & Sons, J., on the Strength of
WroughtIron Tubes, 692, 693.
Ryland Brothers, Warrington Wire Gauge by,
133. 247
Sauvage, on Charcoal, 447, 449, 452.
ScheurerKestner & Meunier  DoUfus, on
French and other Coals, and Lignites, 422,
797
Sharp, Henry, on Rivetted Joints of Steel
Plates, 642.
Shields, F. W., on CastIron Columns, 645.
Shock. Chief Engineer W. H., on Shearing
Strength of Bar Iron, 587.
Siemens, Dr. C. W., on Isolated Steam, 383;
on the Consumption of Fud in Metallurgical
Furnaces, 497; on the Strength of Hot
Blast Iron, 556 ; on HotAir Engines, 920 ;
his Water Pyrometer, 967.
Simms, F. W., on the Work of Horses, 720.
Smeaton. on the Power of Labourers. 718.
Smith, G. Graham, on Strength of Timber,
543. 544. 549
Smith, J. T., on Punching Resistance of Steel,
617 ; on the Strength of Rails, 664.
Snelus, G. J., Analysis of Welsh Coal by, 413.
SocUU IndustrielU de Mulhouse: — on Steam
Boilers, 796.
Soctiti IndustrielU Minerale, Bulletin de
la: — M. Comut on CompressedAif Machi
nery, 896 ; M. Mallard on CompressedAir
Machines, 902.
Sociiti des InginUurs Civils, Comptes Rendus
de la: — Anemometer by M. Arson. 892.
SociH6 Vaudoise des Inginieurs et des Archi
tecies, Bulletin de la: — M. Piccard on
Compressed Air, 911.
Society of Arts, Committee of, on Resistance
on Common Roads, 963.
Society of Arts, Journal of: — on Resistance
on Common Roads, 963.
Spill, Strength of Belting by, 680.
Steel Committee of Civil Engineers, on the
Strength of Wrought Iron, 579, 580 ; and of
Steel, 596603,
Stephenson, Robert, on the Strength of Cast
Iron, 555, 561.
Stoney, on Stress in a Curved Flange, 525;
on Sectional Area of a Continuous Web,
526; on Shearing Strength of Cast Iron,
561 ; his Factors of Safety, 625 ; on the Re
sistance of Columns, 643, 645, 646; on
Stresses in Roofs, 715.
Sullivan, Dr., on Peat, 207.
SutclifTe, on Condensation of Steam in the
Cylinder, 880.
Swindell, J. S. E,, on Ventilation of Mines,
935
Sylvester, Cockle Stove by, 488.
Tangye, J., on the Compressive Resistance of
Wrought Iron, 582.
Tasker, Work of Steam in Portable Engine
by, 883.
Telford, Thomas, on the Strength of Wrought
Iron, 567 ; and of Iron Wire, 586.
Thomas & Laurens, on Brown Charcoal, 449;
on Heating by Steam, 463, 468.
Thomson, David, on Expansive Action of
Steam, 822, 882; on Centrifugal Pumps,
946 ; Duty of Pumping Engines, 948.
Thomson, Professor James, Vortex Wheel
by. 943
Thurston, on the Strength of Iron Wire, 587.
Thwaites & Carbutt, on Root's Blower, 928.
Towne, H. R., on Leather Belts, 679, 742,
745. 748750
Tredgold, Weight and Volume of Various
Substances by, 213 ; on Cooling Hot Water,
469; on Cooling of Steam in Pipes, 472,
474 ; on the Work of a Horse, 720.
Tresca, on Laubereau's HotAir Engine,
917; on GasEngines, 920. 921, 923; on
Pumps, 945, 946 ; on Resistance of Tram
way Omnibus, 961.
Turner, Work of Steam in Portable Engine
by, 883.
Tweddell, R. H., on Shafting, 763.
U
Umber, on M. Him's Wire Ropes. 754.
Unwin, on Strength of Columns, 645.
Ure, Specific Gravity of Alloys by, 20a
AUTHORITIES CONSULTED OR QUOTED.
XXVll
Vidcers, T. K., on the Strength of Steel, 621,
622.
\'»lette. on Wood, 439, 441, 44a, 445; on
Charcoal, 446448. 450, 451
W
Wade, Major, on the Strength of Cast Iron,
557
Walker, B., on Resistance of Shafting, 766.
Walker. John, on the Work of Labourers,
718.
Webb, F. "NV.. on the Strength of Steel, 614,
621.
Vfdtibcr, S.. on Mill Shafting. 763. 764, 766.
Westmacott, Percy, on Shafting, 766; on
Com  Warehousing Machinery, 961; on
Axuibt i o og's Hydraulic Machines, 950.
WhiteUw, James, Water Mill by, 939.
Whitworth, Sir Joseph, Standard WifeGauge
by, 133, 134; Strength of his FluidCom
pressed Steel and of Iron, 614, 615; on
Resistance of Steel and Iron to Explosive
Force, 622; his System of Standard Sizes
of Bolts and Nuts, 681 ; Standard Pitches
of ScrewedIron Piping, 683.
Wiesbach, Coefl&cients for Flow of Water,
892.
Williams, R; Price, on the Transverse Strength
of Rails, 662, 664.
Williams, Foster, & Co., Weight of Sheet
Copper by, 261.
Wilson, A., on the Work of Bullocks, 720.
Wilson, R., on Sttength of Perforated Iron
Plates, 633.
Wilson, Robert (Patricroft), on Teeth of
Wheels, 732.
Wood, J. & E., Work of Steam in Stationary
Engine by, 882.
Woods, E., and J. Dewrance, on the Efficiency
of Heating Surface of a Locomotive, 803.
Wright, J. G., on.Rivetted Joints, 637.
A MANUAL
OF
RULES, TABLES, AND DATA
FOR
MECHANICAL ENGINEERS.
GEOMETRICAL PROBLEMS.
PROBLEMS ON STRAIGHT LINES.
Problem I. — To bisect a straight
iine^ or an arc of a circle^ Fig. i. —
From the ends a, b, as centres, de
scribe arcs intersecting at c and d,
and draw c d, which bisects the line,
or the arc, at the point e or f.
Problem IL — To draw a perpen
dicuiar to a straight line^ or a radial
line to a circtilar arCy Fig. i. — Operate
:c
^
.4.
:b :
:d *•
F^5. I. — Probs. L and II.
as in the foregoing problem. The
line CD is perpendicular to a b : the
line c D is also radial to the arc a b.
Problem IIL — To draw a perpen
dicular to a straight line, from a given
point in that iinc,Yig. 2. — With any
radius, from the given point a, in the
line B c, cut the line at b and c; with
a longer radius describe arcs from b
Fig. 2.— Prob. III.
and c, cutting each other at d, and
draw the perpendicular d a.
2d Method, Fig. 3. — ^rom any cen
tre F, above bc, describe a circle
passing through the given point a,
Fig. 3.— Prob. III. 3d method.
and cutting the given line at d; draw
D F, and produce it to cut the circle
at e; and draw the perpendicular a e.
GEOMETRICAL PROBLEMS
3// Method^ Fig. 4. — From a de
scribe an arc eg, and from e, with
the same radius, the arc a c, cutting
/
'/"\
Fig. 4, — Prob. III. 3d method.
the Other at c ; through c draw a Une
E c D, and set off c d equal to c e ;
and through d draw the perpendicu
lar AD.
4//z Method, Fig. 5. — From the
given point A set off a distance a e
••c
m
^'
» ••
.. 4
E'
Fig. 5.— Prob. III. 4th method.
equal to three pjrrts, by any scale;
and on the centres a and e, With
radii of four and five parts respec
tively, describe arcs intersecting at c.
Draw the perpendicular a c.
Note, — This method is most useful
on very large scales, where straight
edges are inapplicable. Any multi
ples of the numbers 3, 4, 5 may be
taken with the same effect, as 6, 8,
10, or 9, 12, 15.
Problem IV. — To draw a perpen
diadar to a straigJit line from any
point without it, Fig. 6. — From the
point A, with a sufficient radius, cut
the given line at rand g; and frt)m
these points describe arcs cutting at
e. Draw the perpendicular a e.
Note. — If there be no room below
the line, the intersection may be taken
above the line; that is to say, be
tween the line and the given point.
Fig. 6.Prob. IV.
2d Method, Fig. 7. — From any two
points B, c, at some distance apart,
F
/
B
Fig. 7.— Prob. IV. ad method.
in the given line, and with the radii
B A, c A, respectively, describe arcs
cutting at a d. Draw the perpendi
cular A D.
Problem V. — 72? draw a straight
line parallel to a giveti line, at a giten
distance apart. Fig. 8. — From the cen
c
D
,.
^ ...
1 ....
i
•
B
Fig. 8.— Prob. V.
tres A, B, in the given line, with the
given distance as radius, describe arcs
c, D ; and draw the parallel line c D
touching the arcs.
Problem VI. — To draw a parallel
through a given point. Fig. 9. — With
a radius equal to the distance Of the
ON STRAIGHT LINES.
given point c from the given line
A B, describe the arc d from b, taken
1
.^ ■.
A
rig.
9
B
Prob. VI.
considerably distant from c Draw
the parallel through c to touch the
arc D.
2d Method^ Fig. lo. — From a, the
— !p
•
I
_t
Fig. lo.— Prob. VI. ad method.
given point, describe the arc f d, cut
ting the given line at f; from f, with
the same radius, describe the arc e a,
and set oflf f d equal to e a Draw
the parallel through the points a, d.
Note, Fig. II. — When a series of
parallels are required perpendicular
to a base line a b, they may be drawn,
as in Fig. i, through points in the
base line, set oflf at the required dis
t
\
*
•B
Fig. XX.— Prob. VI.
tamces apart This method is con
venient also where a succession of
parallels are required to a given line,
c D ; for the perpendicular a b may be
drawn to it, and any number of par
allels may be drawn upon the per
pendicular.
Problem VII. — To divide a straight
line into a number of equal parts ^ Fig.
12. — To divide the line a b into, say,
five parts. From a and b draw par
allels A c, B D, on opposite sides. Set
off any convenient distance four times
>'
A^
X — %
X
V
^
Fig. 13.— Prob. VII.
(one less than the given number)
from A on AC, and from b on bd;
join the first on a c to the fourth on
B D, and so on. The lines so drawn
divide a b as required.
2d Method^ Fig. 13. — Draw the line
A c at an angle from a, set off, say.
h % 9 A
Fig. 13.— Prob. VII. ad method.
five equal parts; draw b 5, and draw
parallels to it from the other points
of division in a c. These parallels
divide a b as required.
Note. — By a similar process a line
may be divided into a number of
unequal parts; setting oflf divisions
on A c, proportional by a scale to the
required divisions, and drawing par
allels cutting A 6.
Problem VIII. — Upon a straight
GEOMETRICAL PROBLEMS
line to draw an angle equal to a given
angle. Fig. 14. — I^t a be the given
angle, and fg the line. With any
radius, from the points a and f, de
scribe arcs D E, I H, cutting the sides
of the angle a, and the line f g. Set
Fig. 14.— Prob. VIII.
ofl the arc i h equal to d e, and draw
F H. The angle f is equal to a, as
required.
To draw angles of 60** and 30°, Fig.
1 5. — From F, with any radius f i, de
scribe an arc i h ; and from i, with
the same radius, cut the arc at h, and
.H^.
F K I
Fig. 15.— Prob. VIII.
draw F H to form the required angle
I F H. Draw the perpendicular h k
to the base line, to form the angle of
30*" F H K.
To draw an angle of 45**, Fig. 16.
— Set off the distance f i, draw the
i.r I
Fig. 16.— Prob. VIII.
perpendicular i h equal to i f, and
join H F, to form the angle at f as re
quired. The angle at h is also 45**.
Problem IX. — To bisect an angle,
Fig. 17. — Let A c B be the angle; on
the centre c cut the sides at a, b. On
A and B, as centres, describe arcs
cutting at d. Draw c d, dividing the
angle into two equal parts.
Fig. 17. — Prob. IX.
Problem X. — To bisect the inclina
tion of two lines, of which the intersec
tion is ifiaccessible, Fig. 18. — Upon the
B
Fig. 18.— Prob. X.
given lines cb, ch, at any points,
draw perpendiculars e ^, g h, of equal
lengths, and through f and g draw
parallels to the respective lines, cut
ting at s; bisect the angle fsg, so
formed, by the line s d, which divides
equally the inclination of the given
lines.
ON STRAIGHT LINES AND CIRCLES.
5
PROBLEMS ON STRAIGHT LINES
AND CIRCLES.
Problem XI. — Through tuto given
points to describe an arc of a circle with
a given radius^ Fig. 19. — On the points
Problem XIIL — To describe a cir
de passing through three given points^
Fig. 21. — Let A, B, c be the given
points, and proceed as in last pro
Fig. 19.— Proh. XI.
A and B as centres, with the given
radios, describe arcs cutting at c; and
from c, with the same radius, describe
an arc a b as required.
Problem XII. — To find the centre
of a cirde^ or of an arc of a drcle,
I St, for a circle, Fig. 20. — Draw the
Fig. 21.— Prob. XII. XIII.
blem to find the centre o, from which
the circle may be described.
Note, — ^This problem is variously
useful: — in striking out the circular
arches of bridges upon centerings,
when the span and rise are given;
describing shallow pans, or dished
iM
Fig. aa— Prob. XII.
chord A b, bisect it by the perpendi
cular c D, bounded both ways by the
circle; and bisect c d for the centre g.
2d, for a circle or an arc, Fig. 21.
— ^Select three points, a, b, c, in the
circumference, well apart; with the
same radius, describe 2C£t!& from these
three points, cutting each other; and
draw the two lines, d e, f g, through
their intersections, according to Fig. i.
The point o, where they cut, is the
centre of the circle or arc.
Fig. 82.— Prob. XIV. xst method.
covers of vessels ; or finding the dia
meter of a flywheel or any other
object of large diameter, when only
a part of the circumference is ac
cessible.
Problem XIV. — To describe a drcle
passing through three given points when
the centre is not available,
1st Mdhod, Fig. 22. — From the
extreme points a, b, as centres, de
scribe arcs AH, BG. Through the
third point c, draw a e, b f, cutting
GEOMETRICAL PROBLEMS
the arcs. Divide a f and b e into
any number of equal parts, and set
off a series of equal parts of the same
length on the upper portions of the
arcs beyond the points e, f. Draw
straight lines, b l, b m, &c., to the divi
sions in A f; and a i, a k, &c., to the
divisions in e g ; the successive inter
sections N, o, &c., of these lines, are
points in the circle required, between
the given points a and c, which may
be filled in accordingly: similarly the
remaining part of the curve b c may
be described.
2d Methody Fig. 23. — Let a, d,b be
the given points. Draw a b, a d, d b,
Fig. 23. — Prob. XIV. 2d method.
and ef parallel to a b. Divide d a
into a number of equal parts at i, 2, 3,
&c., and from d describe arcs through
these points to meet ef. Divide the
arc A e into the same number of equal
parts, and draw straight lines from d
to the points of division. The inter
sections of these lines successively
with the arcs i, 2, 3, &c., are points
in the circle which may be filled in as
before.
Note. — ^The second method is not
perfectly exact, but is sufficiently near
to exactness for arcs less than one
fourth of a circle. When the middle
point is equally distant fiom the ex
tremes, the vertical c d is the rise of
the arc; and this problem is service
able for setting circular arcs of large
radius, as for bridges of very great
Fig. 24.— Prob. XV.
span, when the centre is unavailable;
and for the outlines of bridgebeams,
and of beams and connectingrods of
steamengines, and the like.
Problem XV. — To draw a tangent
to a circle from a given point in the
circumference^ Fig. 24. — Through the
given point a, draw the radial line
— E
Fig. 25. — Prob. XV. 2d method.
A c, and the perpendicular f g is the
tangent
2d Method^ when the centre is not
available, Fig. 25. — From a, set off
equal segments a b, a d; join b d, and
draw A e parallel to it for the tangent
Problem XVI. — To draw tangents
to a circle from a point without it.
Fig. 36.— Prob. XVI. ist method.
\5t Method^ Fig. 26. — Draw ac
from the given point a to the centre
ON STRAIGHT LINES AND CIRCLES.
G; bisect it at d, and from the centre
D, describe an arc through c, cutting
the circle at e, f. Then a e, a f, are
tangents.
id Method^ Fig. 27. — ^From a, with
the radius a c, describe an arc b c d,
and from c, with a radius equal to the
•.\i>
Fig. vj. — Prob. XVI. 2d method.
diameter of the circle, cut the arc at
B, D ; join B c, CD, cutting the circle
at E,F, and draw ae, af, the tan
gents.
Note. — ^When a tangent is already
drawn, the exact point of contact
may be found by drawing a perpen
dicular to it from the centre.
Problem XVIL — Between two in
clined lines to draw a series of circles
touching these lines and touching ecuh
ather^ Fig. 28. — Bisect the inclination
Fig. a8.— Prob. XVII.
of the given lines a b, c d by the line
NO. From a point p in this line,
draw the perpendicular p b to the line
A B, and on p describe the circle b d
touching the lines and cutting the
centre line at e. From e draw e f
perpendicular to the centre line, cut
ting A B at F, and from f describe an
arc E G, cutting a b at g. Draw g h
parallel to b p, giving h, the centre
of the next circle, to be described
with the radius h e, and so on for the
next circle i n.
Inversely, the largest circle may
be described first, and the smaller
ones in succession.
Note, — This problem is of frequent
use in scroll work.
Problem y>N\\l,^^ Between two
inclined lines to draw a circular seg
ment to fill the angle, and touching the
lines, Fig. 29. — Bisect the inclination
Fig. 99. Prob. XVIII.
of the lines a b, d e by the line f c,
and draw the perpendicular a f d to
define the limit within which the cir
cle is to be drawn. Bisect the angles
a and D by lines cutting at c, and
from c, with radius c f, draw the arc
H F G as required.
Problem XIX. — To describe a cir
cular arc joining two circles, and to
touch one of tliem at a given point. Fig.
30. — ^To join the circles a b, f g, by
an arc touching one of them at f,
draw the radius e f, and produce it
both ways; set off fh equal to the
radius ac of the other circle, join ch
8
GEOMETRICAL PROBLEMS
and bisect it with the perpendicular
L I, cutting E F at i. On the centre i,
Fig. 30. — Prob. XIX.
With radius i f, describe the arc f a as
required.
PROBLEMS ON CIRCLES AND
RECTILINEAL FIGURES.
Problem XX. — To construct a tri
angle on a given base, the sides being
given,
I St. An equilateral triangle, Fig. 31.
Fig. 31.— Prob. XX.
— On the ends of the given base, a, b,
with A B as radius, describe arcs cut
ting at c, and draw a c, c b.
2d. A triangle of unequal sides,
Fig. 32. — On either end of the base
A D, with the side b as radius, describe
an arc ; and with the side c as radius,
on the other end of the base as a
centre, cut the arc at e. Join a e, d e.
Note, — This construction may be
used for finding the position of a
point, c or e, at given distances from
the ends of a base, not necessarily to
form a triangle.
A
Fig. 32.Prob. XX.
Problem XXI. — To construct a
square or a rectangle onagiveti straight
line,
I St. A square. Fig. 33. — On the
Fig. 33. — Prob. XXI.
ends A, b, as centres, with the line a b
as radius, describe arcs cutting at c;
on c, describe arcs cutting the others
at D E ; and on d and e, cut these at
F G. Draw A F, B G, and join the in
tersections H, L
2d. A rectangle. Fig. 34. — On the
base E F, draw the perpendiculars e h,
K
Fig. 34.Prob. XXL
F G, equal to the height of the rect
angle, and join g h.
When the centre lines, a b, c d,
Fig. 35, of a square or a rectangle are
given, cutting at e. — Set off e f, eg,
ON CIRCLES AND RECTILINEAL FIGURES.
the half lengths of the figure, and e h,
ET, the half heights. On the centres
H, T, with a radius of half the length,
•B
1^
A:?'
B
Fig 35.— Prob. XXI.
describe arcs; and, on the centres F,
G, with a radius of half the height, cut
these arcs at k, l, m, n. Join these
intersections.
Problem XXII. — 7b construct a
parallelogram^ of which the sides and
one of the angles are given, Fig. 36. —
CiLLl
B
Fig. 36.— Prob. XXII.
Ehaw the side d e equal to the given
length A, and set off the other side
D F equal to the other length b, form
ing the given angle c. From e, with
D F as radius, describe an arc, and
from F, with the radius d e, cut the
arc at G. Draw f o, eg.
Or, the remaining sides may be
drawn as )arallels to d e, d f.
The formation of the angle d is
readily done as indicated, by taking
the straight length of the arc h i and
CI as radius, and finding the inter
section L.
Problem XXIII. — To describe a
circle about a triangle, Fig. 37. — Bisect
two sides a b, a c of the triangle at
E, F, and from these points draw per
pendiculars cutting at k. On the
centre k, with the radius K a, draw
the circle a b c.
Fig. 37.— Prob. XXI 1 1.
Problem XXIV. — To inscribe a
circle in a triangle. Fig. 38. — Bisect
two of the angles a, c, of the triangle
by lines cutting at d; from d draw a
perpendicular d e to any side, and
with D E as radius describe a circle.
When the triangle is equilateral,
the centre of the circle may be found
by bisecting two of the sides, and
Fig. 38.— Prob. XXIV.
drawing perpendiculars as in the pre
vious problem. Or, draw a perpen
dicular from one of the angles to the
opposite side, and from the side set
off onethird of the perpendicular.
Fig. 39.— Prob. XXV.
Problem XXV. — To describe a
circle about a square, and to inscribe
a square in a circle, Fig. 39.
10
GEOMETRICAL PROBLEMS
I St. To describe the circle. Draw
the diagonals a b, c d of the square,
cutting at e; on the centre e, with the
radius e a, describe the circle.
2d. To inscribe the square. — Draw
the two diameters a b, c d at right
angles, and join the points a, b, c, d
to form the square.
Note, — In the same way a circle
may be described about a rectangle.
Problem XXVL — To inscribe a
circle in a square^ and to describe a
square about a circle^ Fig. 40.
ist. To inscribe the circle. — Draw
Fig. 4o.^Prob. XXVI.
the diagonals a b, c d of the square,
cutting at e; draw the perpendicular
e F to one side, and with the radius
E F describe the circle.
2d. To describe the square. — Draw
two diameters a b, c d at right angles,
and produce them; bisect the angle
D E b at the centre by the diameter
F G, and through f and g draw per
Fig. 41.— Prob. XXVII.
pendiculars ac, bd, and join the
points A D and b c, where they cut the
diagonals, to complete the square.
Problem XXVIL — To inscribe a
pentagon in a circle^ Fig. 41. — Draw
two diameters a c, b d at right angles,
cutting at o; bisect a o at e, and from
E, with radius e b, cut a c at f ; from b,
with radius b f, cut the circumference
at G, H, and with the same radius step
round the circle to i and k; join the
points so found to form the pentagon.
Problem XXVIIL — To construct
a hexagon upon a given straight line^
Fig. 42. — From a and b, the ends of
the given line, describe arcs cutting
at^; from^, with tlie radius^ a, de
^ B
Fig. 4a.Prob. XXVIII.
scribe a circle; with the same radius
set off the arcs a g, g f, and b d, d e.
Join the points so found to form the
hexagon.
Problem XXIX. ^72? inscribe a
hexagon in a circle^ Fig. 43. — Draw a
diameter AC B; from a and b as centres,
with the radius of the circle a c, cut
the circumference at d, e, f, g; and
draw AD, D E, &c. to form the hexagon.
o ^9
Fig. 43.Prob. XXIX
The points d, e, &c., may also be
found by stepping the radius six
times round the circle.
"N
ON CIRCLES AND RECTILINEAL FIGURES.
II
Problem XXX. — To describe a hex
a^m aJbaut a circle^ Fig. 44. — Draw a
a *— .
F7g. 44. — Prob. XXX.
diameter adb, and with the radius
A D, on the centre a, cut the circum
ference at c; join ac, and bisect it
with the radius d e; through £ draw
the parallel f g cutting the diameter
at F, and with the radius d f describe
the circle f h. Within this circle de
scribe a hexagon by the preceding
problem; it touches the given circle.
Problem XXXI, — To describe an
octagon on a given straight line^ Fig. 45.
A.
Fig. 45
B
Prob. XXXI.
— Produce the given line ab both
ways, and draw perpendiculars ae,
BF; bisect the external angles a and
B, by the lines ah, b c, which make
equal to a b. Draw c D and H g par
allel to A E, and equal to a b ; from the
centres g, d, with the radius a b, cut
the perpendiculars at e, f, and draw
E F to complete the octagon.
Problem XXXIL — To convert a
square into an octagon. Fig. 46. — Draw
the diagonals of the square cutting at
ej from the comers a, b, c, d, with a e
as radius, describe arcs cutting the
Fig. 46.Prob. XXXII.
sides at g, h, &c.; and join the points
so found to form the octagon.
Problem XXXIIL — To inscribe
an octagon in a circle, Fig. 47. — Draw
B
Fig. 47.Prob. XXXIII.
two diameters a c, b d at right angles ;
bisect the arcs ab, bc, &c., at ^,/,
&c., and join a ^, ^b, &c., to form the
octagon.
Problem XXXIV. — To describe
an octagon about a circle, Fig. 48. —
Fig. 48. Prob. XXXIV.
Describe a square about the given
circle .A B, draw perpendiculars hk,
12
GEOMETRICAL PROBLEMS
&c., to the diagonals, touching the
circle, to form die octagon.
Or, the points //, k, &c, may be
found by cutting the sides from the
comers of the square, as in the second
last problem.
Problem XXXV. — To describe a
polygon of any number of sides upon a
given straight Hne, Fig. 49. — Produce
Fig. 49. — Prob. XXXV.
the given line a b, and on a, with
the radius a b, describe a semicircle,
divide the semicircumference into as
many equal parts as there are to be
sides in the polygon; say, in this ex
ample, five sides. Draw lines from
A through the divisional points d, b,
and r, omitting one point a, and on
the centres b, d, with the radius a b,
cut A ^ at E and a r at f. Draw
D E, E F, F B to complete the polygon.
Problem XXXVI. — To inscribe a
circle within a polygon, Figs. 50, 51. —
When the polygon has an even num
ber of sides, Fig. 50, bisect two op
Fig. sa— Prob. XXXVI. XXXVII.
posite sides at a and b, draw a b, and
bisect it at c by a diagonal d e; and
with the radius ca describe the circle.
When the number of sides is odd,
Fig. 51, bisect two of the sides at a
Fig. 51.— Prob. XXXVI. XXXVII.
and B, and draw lines a e, b d to the
opposite angles, intersecting at c;
from c, with the radius c a, describe
the circle.
Problem XXXVI I. — To describe a
circle without a polygon. Figs. 50, 51. —
Find the centre c as before, and \^dth
the radius c d describe the circle.
The foregoing selection of prob
lems on regular figures are the most
usefiil in mechanical practice on that
subject Several other regular figures
may be constructed from them by
bisection of the arcs of the circum
scribing circles. In this way a de
cagon, or tensided polygon, may be
formed from the pentagon, as shown
by the bisection of the arc b h at ^
in Fig. 41. Inversely, an equilateral
triangle may be inscribed by joining
the alternate points of division found
for a hexagon.
Problem XXXVIII.— 7?? inscribe
a polygon of any number of sides
within a circle, Fig. 52. — Draw the
diameter a b, and through the centre
E draw the perpendicular e c, cutting
the circle at f. Divide e f into four
equal parts, and set off three parts
equal to those from f to c. Divide
the diameter a b into as many equal
parts as the polygon is to have sides;
and from c draw c d through the
second point of division, cutting the
circle at d. Then a d is equal to one
ON THE ELLIPSE.
13
side of the polygon, and by stepping
round the circumference with the
Fig. 52.— Proh. XXXVIII.
length A D, the polygon may be com
pleted;
The constructions for inscribing
regular polygons in circles are suit
able also for dividing the circumfer
ence of a circle into a number of
equal parts. To supply a means of
dividing the circumference into any
number of parts, including cases not
provided for in the foregoing prob
lems, the annexed table of angles
relating to polygons, expressed in
dqgrees, will be found of general
utib'ty. In tliis table the angle at
Table of Polygonal Angles.
Number
Angle
Number
Angle
, ofSidcs.
at Centre.
of Sides.
at Centre.
No.
Degrees.
No.
Degrees.
3
120
12
30
4
90
13
27A
5
72
H
25f
6
60
15
24
7
5if
16
22^
8
45
17
«^
9
40
18
20
10
1
36
19
19
II
1
32A
20
18
the centre is found by dividing 360°,
the number of degrees in a circle, by
the number of sides in the polygon ;
and by setting off round the centre
of the circle a succession of angles
by means of the protractor, equal to
the angle in the table due to a given
number of sides, the radii so drawn
will divide die circimiference into the
same number of parts. The triangles
thus formed are termed the elemen
tary triangles of the polygon.
•Problem XXXIX. — To inscribe
any regular polygon in a given circle;
or to divide the circumference into a
^ven number of equal parts, by means
of ihe angle at the centre. Fig. 53. —
Fig. 53.Prob. XXXIX.
Suppose the circle is to contain a
hexagon, or is to be divided at the
circumference into six equal parts.
Find the angle at the centre for a
hexagon, or 60°; draw any radius b c,
and set off, by a protractor or other
wise, the angle at the centre cbd
equal to 60°; then the interval cd is
one side of the figure, 6x segment of
the circumference; and the remaining
points of division maybe found either
by stepping along the circumference
with the distance c d in the dividers,
or by setting off the remaining five
angles, of 60^ each, round the centre.
PROBLEMS ON THE ELLIPSE.
An ellipse is an oval figure, like a
circle in perspective. The line a b,
Fig. 54, that divides it equally in the
direction of its greatest dimension,
is the transverse axis; and the per
pendicular CD, through the centre,
is the conjugate axis. Two points,
F, G, in the transverse axis, are the
14
GEOMETRICAL PROBLEMS
foci of the curve, each being called a
focus; being so placed that the sum
of their distances from either end of
the conjugate axis, c or d, is equal
F'g 54* — Prob. XL.
to the transverse axis. In general, the
sum of their distances from any other
point in the curve is equal to the
transverse axis. A line drawn at
right angles to either axis, and termi
nated by the curve, is a double ordi
nate^ and each half of it is an ordinate.
The segments of an axis between an
ordinate and its vertices are called
abscisses. The double ordinate drawn
through a focus is called the para
ineter of the axis.
The squares of any two ordinates
to the transverse axis, are to each
other as the rectangles of their respec
tive abscisses.
Problem XL. — To describe an el
lipse when the length and breadth are
^veny Fig. 54. — On the centre c, with
A £ as radius, cut the axis a b at f
and G, the foci ; fix a couple of pins
into the axis at f and G, and loop on
a thread or cord upon them equal in
length to the axis a b, so as when
stretched to reach to the extremity c
of the conjugate axis, as shown in dot
lining. Place a pencil or drawpoint
inside the cord, as at h, and guiding
the pencil in this way, keeping the
cord equally in tension, carry the
pencil round the pins f, g, and so
describe the ellipse.
Note, — This method is employed
in setting off elliptical gardenplots,
walks, &c.
2d Method, Fig. 55. — Along the
straight edge of a slip of stiff paper,
mark off a distance a c equal to a c,
Piff 55.— Prob. XL. 2d method.
I
half the. transverse axis; and from
the same point a distance a b equal
to c D, half the conjugate axis. Place
the slip so as to bring the point b on
the line a b of the transverse axis, and
the points on the line de; and set
off on the dra\ving the position of the
point a. Shifting the slip, so that
the point b travels on the transverse
axis, and the point c on the conjugate
axis, any number of points in the
curve may be found, through which
the curve may be traced.
id Metlwd^ Fig. 56. — ^The action
of the preceding method may be em
Fig. 56.— Prob. XL. 3d method.
bodied so as to afford the means of
describing a large curve continuously,
by means of a bar mky with steel
points w, /, k, rivetted into brass slides
adjusted to the length of the semi
axes, and fixed with setscrews. A rec
tangular cross E G, with guiding slots,
is placed coinciding with the two
ON THE ELLIPSE.
IS
axes of the ellipse, ac and bh; by
sliding the points ky /, in the slots, and
carrying round the point m, the curve
may be continuously described. A
pen or pencil may be fixed at m,
4M Method, Fig. 57. — Bisect the
transverse axis at c, and through c
Fig. 57. — Prob. XL. 4th method.
draw the perpendicular d e, making
CD and CE each equal to half the
conjugate axis. From d or e, with
the radius a c, cut the transverse axis
at r, f', for the foci. Divide a c into
a number of parts at the points i, 2,
3, &c. With the radius a i, on f and
f' as centres, describe arcs ; and with
the radius b i, on the same centres,
cut these arcs as shown. Repeat the
operation for the other divisions of
the transverse axis. The series of
intersections thus made are points in
the curve, through which the curve
may be traced.
5/>i Method, Fig. 58.— On the two
^••■•'
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T
of points, a, b, &c., in the circumfer
ence A F B, draw radii cutting the in
ner circle at d, b\ &c. From a, by
&c., draw perpendiculars to ab; and
from a'y ^', &c., draw parallels to a b,
cutting the respective perpendiculars
at «, Oy &c. The intersections are
points in the curve, through which
the curve may be traced.
6/// Metlwdy Fig. 59. — When the
transverse and conjugate diameters
Fig. 58.— Prob. XL. 5th method.
axes A B, D E as diameters, on centre
c, describe circles; from a number
Fig. 59.— Prob. XL. 6ih method.
are given, a b, c d, draw the tangent
E F parallel to a b. Produce c d,
and on the centre c, with the radius
of half a B, describe a semicircle hdk;
from the centre g draw any number
of straight lines to the points e, r,
&c., in the line e f, cutting the cir
cumference at /, w,«, &c.; from the
centre o of the ellipse draw straight
lines to the points e, r, &c., and
from the points /, w, «, &c., draw
parallels to g c, cutting the lines o e,
or, &c., at L, M, N, &C. These are
points in the circumference of the
ellipse, and the curve may be traced
through them. Points in the other
half of the ellipse are formed by ex
tending the intersecting lines as indi
cated in the figure.
Problem XLL — To describe an
ellipse approximately by means of cir
cular arcs. — First, with . arcs of two
radii. Fig. 60. — Find the difference
i6
GEOMETRICAL PROBLEMS
of the two axes, and set it off from the
centre o to a and r, on oa and oc;
Fig^. 60.— IVob. XLI.
draw a c, and set off half ac to d;
draw di parallel to ac^ set off o^
equal to o//, join ei, and draw the
parallels e m, d m. From m, with
radius m c, describe an arc through
c; and from / describe an arc through
D ; from d and e describe arcs through
A and B. The four arcs form the
ellipse approximately.
Note. — ^This method does not ap
ply satisfactorily when the conjugate
axis is less than twothirds of the
transverse axis.
o M equal to c l, and on d describe
an arc with radius dm; on a, with
radius o l, cut this arc at a. Thus
the five centres d, a, by h, h' are found,
from which the arcs are described to
form the ellipse.
Note, — ^This process works well for
nearly all proportions of ellipses. It
is employed in striking out vaults and
stone bridges.
Problem XLII. — To draw a tan
'K o'.
%^» /
\
V.
Fig. 6x. — Prob. XLI. ad method.
Second, with arcs of three radii,
Fig. 61. — On the transverse axis ab
draw the rectangle b g, on the height
o c ; to the diagonal a c draw the per
pendicular ghd; set off ok equal
to c, and describe a semicircle on
AK, and produce oc to l; set off
Fig. 62. Prob. XLII.
gefit to an ellipse through a given point
in theatrve. Fig. 62. — From the given
point T draw straight lines to the
foci F, f'; produce f t beyond the
curve to c, and bisect the exterior
angle ^ t f, by the line t //, which
is the tangent.
Problem XLIII. — To draw a
tangent to an ellipse from a given
point without the curve. Fig. 63. —
From the given point t, with a
radius to the nearest focus f, de
scribe an arc on the other focus
' f', with a radius equal to the trans
verse axis, cut the arc at k l, and
Fig. 63.Prob. XLIII.
draw K f', L f', cutting the curve at
M, N. The lines t m, t n are tangents.
ON THE PARABOLA.
17
PROBLEMS ON THE PARABOLA.
A parabola, dac, Fig. 64, is a
cun'e such that every point in the
curve is equally distant from the di
rectrix K L and the focus r. The
focus lies in the axis a b drawn from
the v€rtex or head of the curve a, so
as to divide the figure into two equal
parts. The vertex a is equidistant
from the directrix and the focus, or
A^=AF. Any line parallel to the
axis is a diameter. A straight line,
as EG or D c, drawn across the figure
at right angles to the axis is a double
ordinate, and either half of it is ap
ordinate. The ordinate to the axis
E F G, drawn through the focus, is
called iki't parameter of the axis. A
s^ment of the axis, reckoned from the
vertex, is an absciss of the axis; and
it is an absciss of the ordinate drawn
from the base of the absciss. Thus,
A B is an absciss of the ordinate b c.
Abscisses of a parabola are as the
squares of their ordinates.
Problem XLIV. — To describe a
parabola when an absciss and its ordi
nate are given; that is to say, when
the height and breadth are given,
Fig. 64. — Bisect the given ordinate
<
^
J,
a.
J
^
\,
/•
s
\
\
/
\
V
/
3
$ \
:»
J
1
<» c
rig. 64.— Prob. XLIV.
fic at tf ; draw A a, and then a b per
pendicular to it, meeting the axis at
A Set off A if, A F, each equal to b b;
and draw k ^ l perpendicular to the
aiis. Then k l is the directrix and
F is (he focus. Through f and any
number of points, Oy Oy &c., in the
axis, draw double ordinates, n «,
&c. ; and on the centre f, with tlie
radii ¥ e,oe, &c., cut the respective
ordinates at e, g, «, «, &c. The curve
may be traced through these points
as shown.
2d Method; by means of a square
and a cord. Fig. 65. — Place a straight
Fig. 65.— Prob. XLIV. ad method.
edge to the directrix e n, and apply
to it a square leg. Fasten to the
end G, one end of a thread or cord
equal in length to the edge £ g, and
attach the other end to the focus f;
slide the square along the straight
edge, holding the cord taut against
the edge of the square by a draw
point or pencil d, by which the curve
is described.
^d Method; when the height and the
base are given, Fig. 66. — Let a b be
J5
p
A
F
a
J
r
tr
^
V
Of
^
y
r
1
V
I
c
A
f
^
\
^
1.
m
^
\
a
t
^ i
f t
V 1
^ C
PI
r <
1 1
\ 1
? <
i 1
>
Fig. 66.— Prob. XLIV. 3d method.
the given axis, and c d a double ordi
nate or base; to describe a parabola
2
I8
GEOMETRICAL PROBLEMS
of which the vertex passes through a.
Through a draw e f parallel to c d,
and through c and d draw c e and
D F parallel to the axis. Divide b c
and BD into any number of equal
parts, say five, at «, by &c., and divide
c £ and D F into the same number of
parts. Through the points a^ b, c, d
in the base c d, on each side of the
axis, draw perpendiculars, and through
a, by Cy //, in c £ and d f, draw lines to
the vertex a, cutting the perpendicu
lars at eyfygy h. These are points in
the parabola, and the curve cad may
be traced as shown, passing through
them.
PROBLEMS ON THE HYPERBOLA.
The vertices a, b. Fig. 67, of oppo
site hyperbolas, are the heads of the
curves, and are points in their centre
or axial lines. The transverse axis
A B is the distance between the ver
tices, of which the centre c is the
centre. The conjugate axis g h is a
straight line drawn through the centre
at right angles to the transverse axis.
An ordinate f k is a straight line
drawn from any point of the curve
perpendicular to the axis. The seg
ments of the transverse axis a f, b f,
between an ordinate f k and the ver
tices of the curves, are abscisses.
The parameter is the double ordinate
drawn through the focus. The as
symptotes are two straight lines, s s,
R R, drawn from the centre through
the ends of a tangent ed at the vertex,
equal and parallel to the conjugate
axis, and bisected by the transverse
axis.
The nature of the hyperbola is such
that the difference of the distances
of any point in the curve from the
foci is always the same, and is equal
to the transverse axis.
In a hyperbola the squares of any
two ordinates to the transverse axes
are to each other as the rectangles of
their abscisses.
Problem XLV. — To describe a
hyperbolay the transverse and conjugate
axes being givetiy Fig. 67. — Draw ab
Fig. 67.— Prob. XLV.
equal to the transverse axis, and d e
perpendicular to it and equal to the
conjugate G h. On c, with the radius
c E, describe a circle cutting a b pro
duced, at f/; these points are the foci.
In A B produced take any number of
points Oy Oy &c., with the radii a^^ b^,
and on centres f,/ describe arcs cut
ting each other at «, ;z, &c. These
are points in the curve, through which
it may be traced.
2d Metliody Fig. 67. — The curve
may be drawn thus: — Let the ends
of two threads/p q, f p q, be fastened
at the points /, f, and be made to
pass through a small bead or pin p,
and knotted together at q. Take
hold of Q, and draw the threads tight ;
move the bead along the threads, and
the point ? will describe the curve.
If the end of the long thread be fixed
at F, and the short thread at /, the
opposite curve may be described in
the same ftianner.
Or, the line /q may be replaced
by a straightedge turning on a pin
at/ and the cord F q joined to it at Q.
The curve may then be described by
means of a point or pencil in the same
manner as for the parabola. Fig. 65.
3^/ Method; when the breadth c d.
ON THE HYPERBOLA, CYCLOID, EPICYCLOID.
19
hd^ A B, and transverse axis a a! of
the curve are gtvetiy Fig. 68. — Divide
Fig. 68.— Protx XLV. 3d method.
the base or double ordinate c d into
a number of equal parts on each side
of the axis at a, d, &c ; and divide
the parallels c e, d f, into the same
number of equal parts at a, d, &c.
From the points a, d, &c., in the base,
draw lines to a', and from the points
a, b, &c., in the verticals, draw lines
to A, cutting the respective lines from
the base. Trace the curve through
the intersections thus obtained.
THE CYCLOID AND EPICYCLOID.
Problem XLVL — To describe a
cydoid^ Fig. 69. — When a wheel or a
circle D G c rolls along a straight line
Fig. 69.— Prob. XLVI.
one revolution, it measures off a
straight line a b exactly equal to the
circumference of the circle d g c,
which is called the generating circle,
and a point or pencil fixed at the
point D in the circumference traces
out a curvilinear path a d b, called a
cycloid, A B is the bcLse and c D is the
axis of the cycloid.
Place the generating circle in the
middle of the cycloid, as in the figure,
draw a line e h parallel to the base,
cutting the circle at g; and the tan
gent H I to the curve at the point h.
Then the following are some of the
properties of the cycloid : —
The horizontal line h G=arc of the
circle G d.
The halfbase a c =» the halfcircum
ference c G D.
The arc of the cycloid d h = twice
the chord d g.
The halfarc of the cycloid d a =
twice the diameter of the circle d c.
Or, the whole arc of the cycloid
A D B = four times the axis c d.
The area of the cycloid a d b a =
three times the area of the generating
circle d c
The tangent H i is parallel to the
chord G D.
Problem XLVII. — To describe an
Fig. 7o.Prob. XLVIL
A B, Fig. 69, beginning at a and end exterior epicycloid ^ Fig. 70. — The epicy
ing at D, where it has just completed chid differs from the cycloid in this,
20
GEOMETRICAL PROBLEMS
that it is generated by a point d in
one circle do rolling upon the cir
cumference of another circle a c b,
instead of on a flat surface or line;
the former being the generating circle^
and the latter the fundametital circle.
The generating circle is shown in four
positions, in which the generating
point is successively marked d, d', d",
d'^'. a d"' b is the epicycloid.
Problem XLVIIL — To describe
Fig. 71.— Prob. XLVIIL
an interior epicycloid^ Fig. 71. — If the
generating circle be rolled on the in
side of the fundamental circle, as in
Fig. 71, it forms an interior epicycloid^
or hypocycloidy a j>"' b, which becomes
in this case nearly a straight line. The
other points of reference in the figure
correspond to those in Fig. 70.
When the diameter of the generating
circle is equal to half that of the fun
damental circle, the epicycloid be
comes a straight line, being in fact a
diameter of the larger circle.
THE CATENARY.
«
When a perfecdy flexible string, or
a chain consisting of short links, is
suspended from two points m, n, Fig.
72, it is stretched by its own weight,
and it forms a curve line known as
the catenary, m c n. The point c,
where the catenary is horizontal, is
the vertex.
Problem XLIX. — To describe a
catenary^ Fig. 72. — Draw the vertical
c G equal to the length of the arc of the
chain, m c, on one side of the vertex,
and divide it into a great number of
equal parts,at ( i ), ( 2), (3 ),&c. Draw the
horizontal line c h equal to the length
of so much of the rope or chain as
measures by its weight the horizontal
tension of the chain. From the point
c as the vertex, set off" c (i) on the
horizontal line equal to c i on the
vertical; and (i) (2) from the point
(i), parallel to h i and equal to c(i);
and again (2) (3) from the point (2)
parallel to h 2 and equal to c (i);
and so on till the last segment (6) m
is drawn parallel to h g. The poly
gon c (i) (2) (3) . . . M, thus formed,
is approximately the catenary curve,
which may be traced through the
middle points of the sides of the
polygon. A similar process being
performed for the other side of the
curve, the catenary is completed.
Fig. 7a.— Prob. XLIX.
2d Method. — Suspend a finely
linked chain against a vertical wall.
The curve may be traced from it, on
the wall, ^swering the conditions of
given length and height, or of given
width or length of arc, A cord having
numerous equal weights suspended
from it at short and equal distances
may be used.
CIRCLES, PLANE TRIGONOMETRY.
21
CIRCLES.
The circumference of a circle is commonly signified in mathematical
discussions by the symbol x, which indicates the length of the circumfer
ence when the diameter t= i.
The area of a circle is as the square of the diameter, or the square of the
circumference.
The ratio of the diameter to the circumference is as i to 3*141593 —
commonly abbreviated, as i to 3'i4i6
approximately, as i to 3I
or as 7 to 22
WTien the diameter = i, the area is equal to 785398 +
or, commonly abbreviated, 7854
approximately, j^ths.
HTien the circumference = i, the area is equal to "079577 +
or, abbreviated, , '0796
approximately, A^hs, or '08.
In these ratios, the diameter and the circumference are taken lineally,
and the area superficially. So that if the diameter = i foot, the circum
ference is equal to 3'i4i6 feet, and the area is equal to 7854 square foot
Note, — If the first three odd figures, 1,3, 5, be each put down twice, the
first three of these will be to the last three, that is 113 is to 355, as the
diameter to the circumference.
PLANE TRIGONOMETRY.
The circumference of a circle is supposed to be divided into 360 degrees
or divisions, and as the total angularity about the centre is equal to four right
angles, each right angle contains 90 degrees, or 90°, and half a right angle
<
Fig. 73. — Definidons in Plane Trigonometry.
contains 45**. Each degree is divided into 60 minutes, or 6o'j and, for the
sake of still further minuteness of measurement, each minute is divided into
60 seconds, or 60", In a whole circle there are, therefore, 360 x 60 x 60 =
22 GEOMETRICAL PROBLEMS.
1,296,000 seconds. The annexed diagram, Fig. 73, exemplifies the rela
tive positions of the sine, cosine, versed sine, tangent, cotangent, secant,
and cosecant of an angle. It may be stated, generally, that the correlated
quantities, namely, the cosine, cotangent, and cosecant of an angle, are
the sine, tangent, and secant, respectively, of the complement of the given
angle, the complement being the difference between the given angle and a
right angle. The supplement of an angle is the amount by which it is
less than two right angles.
When the sines and cosines of angles have been calculated (by means of
formulas which it is not necessary here to particularize), the tangents, cotan
gents, secants, and cosecants are deduced from them according to the
following relations : —
rad. X sin. rad/^ rad.*^ rad.^
tan. = ; cotan. = ; sec. = ; cosec. = .
COS. tan. COS. sin.
For these the values will be amplified in tabular form.
A triangle consists of three sides and three angles. When any three
of these are given, including a side, the other three may be found by cal
culation : — ,
Case i. — IVAm a side and its opposite angle are two of the ^vm parts.
Rule i. To find a side, work the following proportion: —
as the sine of the angle opposite the given side
is to the sine of the angle opposite the required side,
so is the given side
to the required side.
Rule 2. To find an angle: —
as the side opposite to the given angle
is to the side opposite to the required angle,
so is the sine of the given angle
to the sine of the required angle.
Rule 3. In a rightangled triangle, when the angles and ofie side tiext t/ie
right angle are given, to find the other side: —
as radius
is to the tangent of the angle adjacent to the given side,
so is this side
to the other side.
Case 2. — When two sides and the included angle are given.
Rule 4. To find the other side: —
as the sum of the two given sides
is to their difference,
so is the tangent of half the sum of their opposite angles
to the tangent of half their difference —
add this half difference to the half sum, to find the greater angle; and
subtract the half difference from the half sum, to find the less angle. Tlie
other side may then be found by Rule i.
Rule 5. When the sides of a rightangled triangle are givai, to find the
angles: — ,
MENSURATION OF SURFACES. 23
as one side
is to the other side,
so is the radius
to the tangent of the angle adjacent to the first side.
Case 3. — When the three sides are given.
Rule 6. To find an angle. Subtract the sum of the logarithms of the
sides which contain the required angle, from 20; to the remainder add the
logarithm of half the sum of the three sides, and that of the difference
bet^s'ccn this half sum and the side opposite to the required angle. Half
the sum of these three logarithms will be the logarithmic cosine of half the
required angle. The other angles may be found by Rule i.
Rule 7. Subtract the sum of the logarithms of the two sides which con
tain the required angle, from 20, and to the remainder add the logarithms
of the differences between these two sides and half the sum of the three
sides. Half the result will be the logarithmic sine of half the required
angle.
Note, — In all ordinary cases either of these rules gives sufficiently accur
ate results. It is recommended that Rule 6 should be used when the
required angle exceeds 90°; and Rule 7 when it is less than 90°.
MENSURATION OF SURFACES.
To find the area of a paraiieiogram. Multiply the length by the height, or
perpendicular breadth.
Or, multiply the product of two contiguous sides by the natural sine
of the included angle.
To find the area of a triangle. Multiply the base by the perpendicular
height, and take half the product
Or, multiply half the product of two contiguous sides by the natural
sine of the included angle.
To find the area of a trapezoid. Multiply half the sum of the parallel
sides by the perpendicular distance between them.
To find the area of a quadrilateral inscribed in a circle. From half the
sum of the four sides subtract each side severally; multiply the four re
mainders together; the square root of the product is the area.
To find the area of any qiiadrilateral figure. Divide the quadrilateral into
two triangles; the sum of the areas of the triangles is the area.
Or, midriply half the product of the two diagonals by the natural sine of
the angle at their intersection.
Note, — As the diagonals of a square and a rhombus intersect at right angles
(the natural sine of which is i), half the product of their diagonals is the
area.
To find the area of any polygon. Divide the polygon into triangles and
trapezoids by drawing diagonals; find the areas of these as above shown,
for the area.
To find the area of a regular polygon. Multiply half the perimeter of the
polygon by the perpendicular drawn from the centre to one of the sides.
Nc^e, — ^To find the perpendicular when the side is given —
24
GEOMETRICAL PROBLEMS.
as radius
to tangent of halfangle at perimeter (see table No. i),
SO is half length of side
to perpendicular.
Or, multiply the square of a side of any regular polygon by the corres
ponding area^in the following table: —
Table No. i. — ^Angles and Areas of Regular Polygons,
Name.
Number
of
Sides.
One half
Angle at the
Perimeter.
Area.
(Side=i)
1
Perpendi
cular.
(Side = I)
Equilateral triangle,
Square,
3
4
5
6
7
8
9
lO
II
12
30°
45°
54°
60°
64°l
67°i
70°
72°
73°A
75°
04330
I oooo
17205
25981
3*6339
48284
6I8I8
76942
93656
III962
02887
05000
06882
08660
1*0383
12071
1*3737
15388
17028
I 8660
.uri^i.M>&^, .•■.«•.
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Undecagon
Dodecagon
To find the circumference of a circle. Multiply the diameter by 3 14 16.
Or, multiply the area by 12*5664; the square root of the product is the
circumference.
To find the diameter of a circle. Divide the circumference by 31416.
Or, multiply the circumference by '3183.
Or, divide the area by 7854; the square root of the quotient is the
diameter.
To find the area of a circle. Multiply the square of the diameter by 7854.
Or, multiply the circumference by onefourth of the diameter.
Or, multiply the square of the circumference by '07958.
To find the length of an arc of a circle. Multiply the number of degrees
in the arc by the radius, and by '01745.
Or, the length may be found nearly, by subtracting the chord of the whole
arc from eight times the chord of half the arc, and taking onethird of the
remainder.
To find the area of a sector of a circle. Multiply half the length of the
arc of the sector by the radius.
Or, multiply the number of degrees in the arc by the square of the radius,
and by 008727.
To find the area of a segment of a circle. Find the area of the sector
which has the same arc as the segment; also the area of the triangle
formed by the radial sides of the sector and the chord of the arc; the
difference or the sum of these areas will be the area of the segment, ac
cording as it is less or greater than a semicircle.
To find the area of a ring included between the circumferences of two con
MENSURATION OF SURFACES.
25
oniric circles. Multiply the sum of the diameters by their difference, and
577854. •
To find Ike area of a cycloid. Multiply the area of the generating circle
To find tJu length of an arc of a parabola^ cut off by a double ordinate
to the axis. To the square of the ordinate add fourfifths of the square of
the absciss; twice the square root of the sum is the length nearly. .
Note. — ^This rule is an approximation which applies to those cases only
in which the absciss does not exceed half the ordinate.
To find the area of a parabola. Multiply the base by the height; two
thirds of the product is the area.
To find the circumference of an ellipse. Multiply the square root of half the
sum of the squares of the two axes by 3*1416.
To find the area of an ellipse. Multiply the product of the two axes by
7^54.
Note, — ^The area of an ellipse is equal to the area of a circle of which
the diameter is a mean proportional between the two axes.
To find the area of an elliptic segment, the base of which is parallel to
either axis of the ellipse. Divide the height of the segment by the axis of
which it is a part, and find the area of a circular segment, by table No.
VII., of which the height is equal to this quotient; multiply the area thus
found by the two axes of the ellipse successively; the product is the area.
To find the length of cm arc of a hyperbola^ beginning at the vertex. To
19 times the transverse axis add 21 times the parameter to this axis, and
multiply the sum by the quotient of the absciss divided by the transverse.
2(L To 9 times the transverse add 2 1 times the parameter, and multiply the
sum by the quotient of the absciss divided by the transverse. 3d. To
each of these products add 15 times the parameter, and then
as the latter sum
is to the former sum,
so is the ordinate
to the length of the arc, nearly.
To find the area of a hyperbola. To the product of the transverse and
absciss add fivesevenths of the square of the absciss, and multiply the square
root of the sum by 21; to this product add 4 times the square root of
the product of the transverse and absciss; multiply the sum by 4 times the
product of the conjugate and absciss, and divide by 75 times the transverse.
The quotient is the area nearly.
To find the area of any cundlinecU figure,
hounded at the ends by parallel straight lines,
Fig. 74. Divide the length of the figure
ab into any even number of equal parts,
and draw oidinates c, d, e, &c, through the
pomts of division, to touch the boundary
lines. Add together the first and last
ordinates {c and k), and call the sum a;
add together the even ordinates (that is,
^Jy ^y ), and call the sum b; add together
the odd ordinates, except the first and last (e,g, i), and call the sum c Let
D be the common distance of the ordinates, then
Fig. 74.— For Area of Curvilinear Figure.
26
GEOMETRICAL PROBLEMS.
(a + 4 B + 2 c)
X D = area of figure.
This IS known as Simpson's Rule.
2d Method^ Fig. 74. — Having divided the figure into an even or an odd
number of equal parts, add together the first and last ordinates, making
the sum a; and add together all the intermediate ordinates, making the
sum B. Let l = the length of the figure, and n = the number of divisions,
then
A + 2B
2n
X L = area of figure.
That is to say, twice the sum of the intermediate ordinates, plus the first
and last ordinates, divided by twice the number of divisions, and multi
plied by the length, is equal to the area of the figure.
This method is that commonly used; it is sufficiently near to exactness
for most purposes.
3^ Methody Fig. 74. — Having divided the' figure as above, measure by a
scale the mean depth of each division, at the middle of the division; add
together the depths of all the divisions, and divide the sum by the number
of divisions, for the average depth; multiply the average depth by the length,
which gives the area.
For the sake of obtaining a more nearly exact result, the figure may be
divided into two halfparts, c^k^ Fig. 75, one at each end, and a number
of whole equal parts, d^e^f^gji^ij^ intermediately. Then the ordinates
separating these parts, excluding the extreme ordinates, may be measured
»
Fig. 75.
For Area of Cunrilincal Figures.
Fig. 76.
direct, and the sum of the measurements divided by the nimiber of them,
and multiplied by the length, for the area.
Note, — In dealing with figures of excessively irregular outline, as in Fig.
76, representing an indicatordiagram from a steamengine, mean Hnes, ab^
c dy may be substituted for the actual lines, being so traced as to intersect
the undulations, so that the total area of the spaces cut off may be com
pensated by that of the extra spaces inclosed.
Note 2. — The figures have been supposed to be bounded at the ends by
parallel planes. But they may be terminated by curves or angles, as in
Fig. 76, at ^, when the extreme ordinates become nothing.
MENSURATION OF SOLIDS. 2^
MENSURATION OF SOLIDS.
To find tke surface of a prism or a cylinder. The perimeter of the end
multiplied by the height gives the upright surface ; add twice the area of
an end.
Tofijid the cubic contents of a prism or a cylinder. Multiply the area of
the base by the height
To find the surface of a pyramid or a cone. Multiply the perimeter of the
base by half the slant height, and add the area of the base.
To find the aibic contents of a pyramid or a cone. Multiply the area of
the base by onethird of the perpendicular height.
To find the surface of afrustrum of a pyramid or a cone. Multiply the
sum of the perimeters of the ends by half the slant height, and add the
areas of the ends.
To find the cubic cofitents of a frustrum of a pyramid, or a cone, — Add
together the areas of the two ends, and the mean proportional between them
(that is, the square root of their product), and multiply the sum by onethird
of the perp>endicular height.
Or, when the ends are circles, add together the square of each diameter,
and the product of the diameters, and multiply the sum by 7854, and by
onethird of the height
To find the aibic contaits of a wedge, — To twice the length of the base
add the length of the edge ; multiply the sum by the breadth of the base,
and by onesixth of the height
To find the cubic contents of aprismoid {a solid of which the tivo etuis are dis
similar but parallel plane figures of the same number of sides), — To the sum
of the areas of the two ends, add four times the area of a section parallel to
and equally distant from both ends; and multiply the sum by onesixth of
the length
Note. — ^This rule gives the true content of all fmstrums, and of all solids
of which the parallel sections are similar figures; and is a good approxima
tion for other kinds of areas and solidities.
To find the surface of a sphere. — Multiply the square of the diameter by
31416.
Note. — ^The surface of a sphere is equal to 4 times the area of one of its
great circles.
2. The surface of a sphere is equal to the convex surface of its circum
scribing cylinder.
3. The surfaces of spheres are to one another as the squares of their
diameters.
To find the curve surface of any segmmt or zone of a sphere. — Multiply the
diameter of the sphere by the height of the zone or segment, and by 3*1416.
Note. — The curve surfaces of segments or zones of the same sphere are
to one another as their heights.
To find the cubic contents of a sphere, — Multiply the cube of the diameter
by 5236.
Or, multiply the surface by onesixth of the diameter.
28 GEOMETRICAL PROBLEMS.
Note. — The contents of a sphere are twothirds of the contents of its
circumscribing cylinder.
2. The contents of spheres are to one another as the cubes of their
diameters.
To find the aibic contents of a segment of a sphere. — From 3 times the
diameter of the sphere subtract twice the height of the segment; multiply
the difference by the square of the height, and by '5236.
Or, to 3 times the square of the radius of the base of the segment, add
the square of its height; and multiply the sum by the height, and by '5236.
To find the cubic contents of a frustrum or zone of a sphere, — ^To the sum
of the squares of the radii of the ends add Yi of the square of the height;
multiply the sum by the height, and by 1*5708.
To find the cubic contents of a spheroid. — Multiply the square of the re
volving axis by the fixed axis and by '5236.
Note. — ^The contents of a spheroid are twothirds of the contents of its
circumscribing cylinder.
2. If the fixed and revolving axes of an oblate spheroid be equal to the
revolving and fixed axes of an oblong spheroid respectively, the contents of
the oblate are to those of the oblong spheroid as the greater to the less axis.
To find the cubic contents of a segment of a spheroid. — ist. When the base
is parallel to the revolving axis. Multiply the difference between thrice the
fixed axis and double the height of the segment, by the square of the height,
and the product by '5236. Then,
as the square of the fixed axis
is to the square of the revolving axis,
so is the last product
to the content of the segment
2d. When the base is perpendicular to the revolving axis. Multiply the
diflference between thrice the revolving axis and double the height of the
segment, by the square of the height, and the product by '5236. Then,
as the revolving axis
is to the fixed axis,
so is the last product
to the content of the segment.
To find tlie solidity of the middle frustrum of a spheroid. — ist When the
ends are circular, or parallel to the revolving axis. To twice the square of
the middle diameter, add the square of the diameter of one end; multiply
the sum by the length of the frustrum, and the product by '2618 for the
content.
2d. When the ends are elliptical, or perpendicular to the revolving axis.
To twice the product of the transverse and conjugate diameters of the middle
section, add the product of the transverse and conjugate diameters of one
end; multiply the sum by the length of the frustrum, and by '2618 for the
content.
To find tJie cubic contents of a parabolic conoid. — Multiply the area of the
base by half the height.
Or, multiply the square of the diameter of the base by the height, and
by 3927.
To find the cubic contents of a frustrum of a parabolic cofioid. — Multiply
half the sum of the areas of the two ends by the height of the frustrum.
MENSURATION OF SOLIDS.
29
Or, muUiply the sum of the squares of the diameters of the two ends by
the height, and by '3927.
Tofifid the cubic contents of a parabolic spindle, — Multiply the square of the
middle diameter by the length, and by 41888.
To find the cubic contents of the middle frustrum of a parabolic spindle, —
Add together 8 times the square of the largest diameter, 3 times the square
of the diameter at the ends, and 4 times the product of the diameters;
multiply the sum by the length of the frustrum, and by '05236.
To find the surface and the cubic contmts of any of the five regular solids^ Figs.
Fig 77
Fig. 78.
Fig. 79
Fig. 80.
Fig. 81.
77, 78, 79, 80, 81. — For the surface, multiply the tabular area below, by the
square of the edge of the solid.
For the contents, multiply the tabular contents below, by the cube of the
given edge.
Note. — ^A regular solid is bounded by similar and regular plane figures.
There are five regular solids, shown by Figs. 77 to 81, namely: —
The tetrahedron^ bounded by four equilateral triangles.
The hexahedron, or cube, bounded by six squares.
The octahedron^ bounded by eight equilateral triangles.
The dadecahedron,\>o\ixidtd by twelve pentagons.
The icosahedron^ bounded by twenty equilateral triangles.
Regular solids may be circumscribed by spheres; and spheres may be
inscribed in regular solids.
Surfaces and Cubic Contents of Regular Solids.
Number
of sides.
Name.
Area.
Edge = I.
Contents.
Edge=i.
4
6
8
12
20
Tetrahedron
Hexahedron
Octahedron '. .
Dodecahedron
Icosahedron
1:7320
6*0000
3*4641
20*6458
86603
0*1178
I *oooo
0*4714
7*6631
2*1817
To find the cubic contents of an irregular solid, — Suppose it divided into
parts, resembling prisms or other bodies measurable by preceding rules; find
the content of each part; the sum of the contents is the cubic contents of
the solid.
Note, — The content of a small part is found nearly by multiplying half
the sum of the areas of each end by the perpendicular distance between
them.
30 GEOMETRICAL PROBLEMS.
Or, the contents of small irregular solids may sometimes be found by im
mersing them under water in a prismatic or cylindrical vessel, and observing
the amount by which the level of the water descends when the solid is
withdrawn. The sectional area of the vessel being multiplied by the descent
of the level, gives the cubic contents.
Or, when the solid is very large, and a great degree of accuracy is not
requisite, measure its length, breadth, and depth in several different places,
and take the mean of the measurement for each dimension, and multiply
the three means together.
Or, when the surface of the solid is very extensive, it is better to divide
it into triangles, to find the area of each triangle,' and to multiply it by the
mean depth of the triangle for the contents of each triangular portion ; the
contents of the triangular sections are to be added together.
The mean depth of a triangular section is obtained by measuring the
depth at each angle, adding . together the three measurements, and taking
onethird of the sum.
MENSURATION OF HEIGHTS AND DISTANCES.
To find the height of an accessible object. — Measure the distance from the
base of the object to any convenient station on the same horizontal plane;
and at this station take the angle of altitude. Then
as radius
to tangent of the angle of altitude,
so is the horizontal distance
to the height of the object above the horizontal plane passing
through the eye of the observer. Add the height of the eye, and the sum
is the height of the object.
Note, — The station should be chosen so that the angle of altitude should
be as near to 45° as practicable; because the nearer to 45'', the less is the
error in altitude arising from error of observation.
When the angle of elevation is 45^ the height above the plane of the
eye is equal to the distance. When it is 26° 34', the height is half the dis
tance.
To find approximately the height of an accessible object. — There are four
methods based on the principle of similar triangles.
I St. By a geometrical square^ Fig. 82. — This is a
square, a b, with two sights on one of its sides, a ;/,
a plumbline hung from one extremity, ;/, of that
side, and each of the twp sides opposite to that
extremity, mb,ma^ divided into 100 equal parts;
the division beginning at the remote ends, so that
the 1 00th divisions meet at the corner m. Let re
be the object, and the sights be directed to the
summit ^, at the known distance ad. When the
Fig. 82.Mcnsuration of a plummet cuts the side b m at, say, c, then by similar
triangles, nb\nc\\ad\de. Or, if the plumbline
cuts the side a m, then the part of a m cut off is to <z « : : ad\ de. Adding
to de the height of the eye rd, the sum is the height of the object, re.
MENSURATION OF HEIGHTS AND DISTANCES.
31
2d. By shadows. Fig. 83.~7When the sun shines, fix a pole ^^ in the
ground, vertically, and measure its shadow a b. Measure also the shadow de
Fig. 83.
Mensuration of a Height.
of the object € m; then, by similar triangles, ab',bc\:de\ e m^ the height of
the object.
3d. By r^4ctiofiy Fig. 84. — Place a basin of water, or any horizontal
reflecting surface, at a, level with the base of the object de, and retire from
it till the eye at c sees the top of the object e,
in the centre of the basin at a. Then, by similar
triangles, abi bc\\ad\de,
4tt By two poles. Fig. 85. — Fix two poles a m,
cfiy of unequal lengths, parallel to the object er,
so that the eye of the observer at a, the top of
the shorter f>ole, may see c, the top of the longer
pole, in a line with e, the summit of the object re.
By similar triangles, ab \bc\\ad\de\ and adding
rd, the height of the eye, to de, the sum r^ is the
height of the object.
To find the distance of the visible horizon. — To
half the logarithm of the height of the eye, add 3*8105; the sum is the
logarithm of the distance in feet, nearly.
To find the distance of an object by the motion of sound, — Multiply the
number of seconds that elapse between the flash or other sign of the gene
ration of the sound and the arrival of the sound to the ear, by 1120. The
product is the distance in feet.
Note. — ^\Vhen a sound generated near the ear returns as an echo, half the
interval of time is to be taken, to find the distance of the reflecting surface.
Fig. 85. Menmration of a
Height.
32 MATHEMATICAL TABLES.
MATHEMATICAL TABLES.
Table No. I. — Of Logarithms of Numbers from i to 10,000.
Logarithms consist of integers and decimals; but, for the sake of com
pactness, the integers have been omitted in the table, except in the short
preliminary section containing the complete logarithms of numbers from i to
100. The table No. I. contains the decimal parts, to six places, of the loga
rithms of numbers from i to 10,000. The integer, or index, or character
istic of a logarithm, standing on the lefthand side of the decimal point, is
a number less by i than the number of figures or places in the integer
of the number. If a number contains both integers and decimals, the
index is regulated according to the integers. If it contain only decimals,
the index is equal to the number of cyphers next the decimal point, plus i;
moreover, the index is negative, and is so distinguished by the sign minus,
— , written over it
For example, to illustrate the adjustment of the integer of the logarithm
to the composition of the number : —
Numbar. Logarithm.
4743 3676053
474.3 2.676053
47.43 L676053
4.743 0.676053
.4743 .L676053
.04743 .£.676053
.004743 3676053
Still more for the sake of compactness, the first two figures of the loga
rithms are given only at the beginning of each line of logarithms, to save
repetition, only the remaining four decimal places being given for each
logarithm. In seeking for a logarithm, the eye readily takes in the prefixed
two digits at the commencement of each line.
Rules, — To find the logarithm of a number containing one or two digits,
look for the number in the preliminary tablet in one of the columns
marked No., and find the logarithm next it Or, look in the body of
the table for the given number in the columns marked N, with one or
two cyphers following it; the decimal part of the logarithm is in the
column next to it For example, the decimal part of the logarithm of 3
is found, in the column next to the number 300, to be .477121, and as
there is but one digit, the logarithm is completed with a cypher, thus,
0.47 7 12 1. The same logarithm stands for 30, except that, when completed,
it becomes 1.477 121. Again, take the number 37; look for 370 in column
N, and the decimal part of the logarithm is found, in the colunm next it,
to be .568202, which, being completed, becomes 1.568202. If the number
be .37, the logarithm becomes 1.568202.
To find the logarithm of a number consisting of three digits, look for the
EXPLANATION AND USES OF THE TABLES. 33
number in column N, and find the logarithm in the column next it, as
already exemplified, for which the index is to be setded and prefixed as
before.
If the number consist of four digits, look for the first three in column N,
and the fourth in the horizontal line at the head or at the foot of the table.
The decimal part of the logarithm is found opposite the three first digits
and under or over the fourth. Take the number 5432; opposite 543 in
column N, and in the column headed 2, is the logarithm .734960, to which
3 is to be prefixed, making 3.734960. If the number be 5.432, the
complete logarithm is 0.734960.
If the number consist of five or more digits, find the logarithm for the
first four as above; multiply the difference, in column D, by the remaining
digits, and divide by 10 if there be only one digit more, by 100 if there
be two more, and so on; add the quotient to the logarithm for the first four.
The sum is the decimal part of the required logarithm, to which the index
is to be prefixed. For example, take 3. 141 6. The logarithm of 3 141 is
.497068, decimal part; and the difference, 138 x 6 h 10 = 83, is to be added,
thus —
0.497068
83
making the complete logarithm, 0.497151
To find the number corresponding to a given logarithm, look for the
logarithm without the index. If it be found exactly or within two or three
units of the righthand digit, then the first three figures of the indicated
number will be found in the number column, in a line with the logarithm,
and the fourth figure at the top or the foot of the column containing the
logarithm. Annex the fourth figure to the first three, and place the decimal
ix)int in its proper position, on the principles already explained.
If the given logarithm differs by more than two or three units from the
nearest in the table, find the number for the next less tabulated logarithm,
which will give the four first digits of the required number. To find the
fifth and sixth digits, subtract the tabulated logarithm from the given loga
rithm, add two c)rphers, and divide by the difference found in column D
opposite the logarithm. Annex the quotient to the four digits already
found,. and place the decimal point For example, to find the number
represented by the logarithm 2.564732: —
2.564732 given logaridim.
Log. 367.0= 2.564666 nearest less.
367.056
56 D 118)6600 (56 nearly.
590
700
708
Showing that the required number is 367.056.
To multiply together two or more numbers, add together the logarithms
. 3
34 MATHEMATICAL TABLES.
of the numbers, and the sum is the logarithm of the product Thus, to
multiply 365 by 3.146: —
Iog 365 = 2.562293
Log 3.146 = 0.497759
3.060052
Log 1148 3059942
29 D 380)11000 (29 nearly.
760
1148.29
3400
3420
Showing that the product is 1148.29.
To divide one number by another, subtract the logarithm of the divisor
from that of the dividend, and the remainder is the logarithm of the
quotient
To find any power of a given number, multiply the logarithm of the num
ber by the exponent of the power. The product is the logarithm of the
power.
To find any root of a given number, divide the logarithm of the number
by the index of the root The quotient is the logarithm of the root
To find the reciprocal of a number, subtract the decimal part of the
logarithm of the number from 0.000000; add i to the index of the loga
rithm, and change the sign of the index. This completes the logarithm of
the reciprocal. For example, to find the reciprocal of 230: —
0.000000
Log 230 = 2.361728
3.638272= log 0.004348 (reciprocal).
Inversely, to find the reciprocal of the decimal .004348 :■—
0.000000
Log .004348 = 3.638272
2.361728 = log 230 (reciprocal).
Note. — It will be found in practice, for the most part, unnecessary to
note the indices of logarithms, as the decimal parts are in most cases suffi
ciently indicative of the numbers without the indices. The exact calcula
tion of differences may also in most cases be dispensed with — rough mental
approximations being sufficiently near for the purpose — particularly when
the numbers contain decimals. The indices are, however, indispensable in
the calculation of the roots of numbers.
EXPLANATION AND USES OF THE TABLES. 35
Tabl^ No. IL — Of Hyperbolic Logarithms of Numbers.
In this table, the numbers range from i.oi to 30, advancing by .01, up
to the whole number 10; and thence by larger intervals up to 30. The
h)'perbolic logarithms of numbers, or Neperian logarithms, as they are
sometimes called, are calculated by multiplying the common logarithms of
the given numbers, in table No.. I., by the constant multiplier, 2.302585.
The hyperbolic logarithms of numbers intermediate between those which
are given in the table, may be readily obtained by interpolating proportional
differences.
Table No. III. — Of Circumferences, Circular Areas, Squares and
Cubes; and of Square Roots and Cube Roots.
It ha«g been shown how to calculate the powers and roots of numbers by
means of logarithms. The table No. III. will be useful for reference. It
contains the powers and roots of numbers consecutively from i to 1000.
The circumferences and areas of circles, due to the numbers contained in
the first columns, considered as diameters, are also given. They will be
found useful when diameters are expressed in integers and decimals, or
otherwise than in feet, inches, and fractions.
Table No. IV. — Of Circumferences and Areas of Circles, with
Sides of Equal Squares.
The Table No. IV. gives the circumferences and areas of circles from
^ inch to 120 inches in diameter, advancing by sixteenths of an inch up
to 6 inches diameter; thence by eighths of an inch to 50 inches diameter;
thence by quarters of an inch to 100 inches diameter; and thence by half
inches to 120 inches diameter.
At the same time, the decimal equivalents of fractions of inches are given
in the first columns, and they are complemented by inches and decimals
advancing by tenths, for which also the circumferences and areas are given.
The table is thus completed for diameters expressed with decimals, as
well as for those expressed with vulgar fractions.
By a suitable adjustment of decimal points the circumferences and areas
may be determined from the contents of the table for diameters ten or a
hundred times as much as, or less than, the values given in the first column.
^\Tiilst the diameters are here expressed as inches, they may be taken as
feet, or as measures of any other denomination.
The column of sides of equal squares^ contains the sides of squares having
the same area as the circles in the same hnes of the table respectively.
Note, — ^The column oi circular areas given in table No. III., contains the
areas of circles of which the diameters are given in common numbers in
the first column.
Tables Nos. V. and VI. — Of Lengths of Circular Arcs.
The lengths of circular arcs are given proportionally to that of the radius,
and to that of the chord, in the tables Nos. V. and VL In the first of these
tables, the radius is taken = i, and the number of degrees in the arc are
given in the first column. The length of the arc as compared with the
radius is given decimally in the second column.
36 MATHEMATICAL TABLES.
In the second table, the chord is taken = i, and the rise or height of the
arc, expressed decimally as compared with the chord, is given in the first
column. The length of the arc relatively to the chord is given in the
second column.
To use the first table, No. V., find the proportional length of the arc
corresponding to the degrees in the arc, and multiply it by the actual length
of the radius; the product is the actual length of the arc.
To use the second table, No. VI., divide the height of the arc by the chord
for the proportional height of the arc, which find in the first column of the
table ; die proportional length of the arc corresponding to it being multi
plied by the actual length of the chord, gives the actual length of the arc.
Note, — The length of an arc of a circle may be found nearly thus: —
Subtract the chord of the whole arc from 8 times the chord of half the arc.
A third of the remainder is the length nearly.
Table No. VII. — Of Areas of Circular Segments.
The areas of circular segments are given in Table No. VII., in proportional
superficial measure, the diameter of the circle of which the segment forms
a portion being = i. The height of the segment, expressed decimally in
proportion to the diameter, is given in the first column, and the relative
area in the second column.
To use the table, divide the height by the diameter, find the quotient in
the table, and multiply the corresponding area by the square of the actual
length of the diameter; the product will be the actual area.
Table No. VIII. — Sines, Cosines, Tangents, Cotangents, Secants,
AND Cosecants of Angles from o° to 90°
This table, Na VIII., is constructed for angles of from 0° to 90°, advancing
by 10', or onesixth of a degree. The length of the radius is equal to i,
and forms the basis for the relative lengths given in the table, and which
are given to six places of decimals. Each entry in the table has a duplicate
significance, being the sine, tangent, or secant of one angle, and at the
same time the cosine, cotangent, or cosecant of its complement For this
reason, and for the sake of compactness, the headings of the columns are
reversed at the foot; so that the upper headings are correct for the angles
named in the left hand mai^n of the table, and the lower headings for
those named in the right hand margin.
To find the sine, or other element, to odd minutes, divide the difference
between the sines, &c., of the two angles greater and less than the given
angle, in the same proportion that the given angle divides the difference of
the two angles, and add one of the parts to the sine next it
By an inverse process the angle may be found for an^ given sine, &c.,
not found in the table.
Table No. IX. — Of Logarithmic Sines, Cosines, Tangents, and Co
tangents of Angles from 0° to 90^
This table. No. IX., is constructed similarly to the table of natural sines,
&c., preceding. To avoid the use of logarithms with negative indices, the
radius is assumed, instead of being equal to i, to be equal to 10", or
EXPLANATION AND USES OF THE TABLES. 37
io,cx)o,ooo,ooo; consequently the logarithm of the radius = lo log lo = lo,
^^Tience, if, to log sine of any angle, when calculated for a radius = i, there
be added lo, the sum will be the log sine of that angle for a radius = lo".
For example, to find the logarithmic sine of the angle 15° 50'.
Nat sine 15'' 50'= 272840; its log = 1*435908
add = 10
Logarithmic sine of 15** 50'= 9*435908
When the logarithmic sines and cosines have been found in this manner,
the logarithmic tangents, cotangents, secants, and cosecants are found from
those by addition or subtraction, according to the correlations of the
trigdnometrical elements already given, and here repeated in logarithmic
fonn: —
Log tan = 10 + log sin.  log. cosin.
Log cotan = 20 — log tan.
Log sec. = 20 — log cosin.
Log cosec = 20  log sin.
To find the logarithmic sine^ tangent^ &*c., of any angle, — ^When the number
of degrees is less than 45°, find the degrees and minutes in the left hand
colunm headed angle^ and under the heading sine^ or tangent^ &c., as
required, the logarithm is found in a line with the angle.
When the number of degrees is above 45°, and less than 90®, find the
degrees and minutes in the right hand column headed angle^ and in the
same line, above the title at the foot of the page, sine or tangent^ &c., find
the logarithm in a line with the angle.
When the number of degrees is between 90° and 180°, take their supple
ment to 180°; when between 180° and 270°, diminish them by 180°; and
when between 270° and 360°, take their complement to 360°, and find the
logarithm of the remainder as before.
If the exact number of minutes is not found in the table, the logarithm
of the nearest tabular angle is to be taken and increased or diminished as
the case may be, by the due proportion of the difference of the logarithms
of the angles greater and less than the given angle.
Table No. X. — Rhumbs, or Points of the Compass.
The Mariner's Compass is a circular card suspended horizontally, having
a thin bar of steel magnetized, — the needle^ — for one of its diameters; the
circumference of the card being divided into 32 equal parts, ox points^ and
each point subdivided into quarters. A point of the compass is, therefore,
equal to (360** ^ 32 = ) n° 15'.
Table No. XL — Of Reciprocals of Numbers.
The table No. XL contains the reciprocals of numbers from i to 1000.
It has already been shown how to find the reciprocal of a number by means
of logarithms.
38
MATHEMATICAL TABLES.
TABLE No. L— LOGARITHMS OF NUMBERS
FROM I TO 10,000.
No.
Log.
1^0.
Log.
No.
Log.
No.
Log.
1
o.oooooo
26
1.414973
51
1.707570
76
1. 880814
2
0.301030
27
1.431364
52
1. 716003
77
1 1. 88649 1
3
0.477I2I
28
. 1.447158
53
1.724276
78
1.892095
4
0.602060
29
1.462398
54
1.732394
79
1.897627
5
0.698970
30
1
1.477121
55
1.740363
80
1.903090
6
0.778I5I
31
1.491362
56
I. 748188
81
1.908485
7
0.845098
32
1.505150
57
1755875
82
1.913814
8
0.903090
33
1.518514
5«
1.76342S
P
1. 919078
9
0954243
34
1.531479
59
1.770852
84
1.924279
lO
I.OOOOOO
35
1.544068
60
1.77S151
85
1.929419
11
I.04I393
36
1556303
61
1.785330
86
1.934498
12
I.079I8I
37
1.568202
62
1.792392
87
1.939519
13
1. 1 13943
38
1.579784
63
I. 799341
88
1.944483
H
I.I46I28
39
1. 591065
64
1. 8061 80
89
I 949390
15
I.I7609I
40
1.602060
65
1.812913
90
91
1.954243
i6
1. 204120
41
1. 612784
66
1819544
1.959041
'Z
1.230449
42
1.623249
^7
1.826075
92
1.963788
i8
1255273
43
1.633468
68
1.832509
93
1.968483
19
1.278754
44
1.643453
69
1.838849
94
1.973128 1
20
I. 301030
45
1. 653213
70
1.845098
1.851258
95
1.977724
21
1. 322219
46
1.662758
71
96
1. 98227 1
22
1.342423
47
1.672098
72
1.857332
97
1.986772
23
1. 361728
48
I.68i24<
73
1.863323
98
1. 991 226
24
1. 38021 1
49
1. 690196
74
1.869232
99
1.995635
25
I 397940
50
1.698970
75
I. 875061
100
2.000000
N
I
2
3
4
5
— 1
6
7
8
9
D
432
100
00 0000
0434
0868
1301
1734
2166
2598
3029
3461
3891
lOI
00 4321
4751
51S1
5609
6038
6466
6894
7321
7748
8174
428
102
00 8600
9026
9451
9876
425
102
01
0300
0724
1 147
1570
1993
2415
424
103
01 2837
3259
3680
4100
4521
4940
5360
5779
6197
6616 1 420
104
01 7033
7451
7868
8284
8700
9116
9532
9947
1 417
104
02
0361
0775 416
105
02 I I 89
1603
2016
2428
2841
3252
3664
4075
4486
4896 412
106
02 5306
5715
6125
6533
6942
7350
7757
8164
8571
8978
408
107
02 9384
9789
405
107
03
0195
0600
1004
1408
1812
2216
2619
3021
404
108
03 3424
3826
4227
4628
5029
5430
5830
6230
6629
7028
400
109
03 7426
7825
8223
8620
9017
9414
981 1
398
109
N
04
0207
0602
8
0998
9
397
I
2
3
4
5
6
7
D
LOGARITHMS OF NUMBERS.
39
N
I
1787
2
2182
3
2576
4
5
6
7
8
9
D
110 i 04 1393
2969
3362
3755
4148
4540
iP^
393
!
'" ' 04 5323
5714
6105
6495
6885
7275
7664
8053
8442
8830
389
112 04 9218
9606
9993
388
112 05
0380
0766
1153
1538
1924
2309
2694
386
"3 05 3078
3463
3846
4230
4613
4996
5378
5760
6142
6524
383
114 1 05 6905
7286
7666
8046
8426
8805
9185
9563
9942
383
114 06
0320
379
1
L15 06 0698
1075
1452
1829
2206
2582
2958
3333
3709
4083
376
116 06 4458
117 06 8186
4832
5206
5580
5953
6326
6699
7071
7443
7815
373
8557
8927
9298
9668
•
380
"7 07
0038
0407
0776
1 145
1514
370
118 07 1882
2250
2617
2985
3352
3718
4085
4451
4816
5182
366
119 07 5547
5912
6276
6640
7004
7368
773»
8094
8457
8819
363
120 1 07 9181
9543
9904
• • * • •
362
120
08
0266
0626
0987
1347
1707
2067
2426
360
121
08 2785
3^44
3503
3861
4219
^576
4934
5291
5647
6004
357
1 122
08 6360
6716
7071
7426
7781
8136
8490
8845
9198
9552
355
: 123 08 9905
355
123 09
0258
0611
0963
'3'5
1667
2018
2370
2721
3071
353
1
124 09 3422
3772
4122
4471
4820
5169
5518
5866
6215
6562
349
125
09 6910
7257
7604
795 »
8298
8644
8990
9335
9681
348
125
10
0026
346
126
10 0371
0715
1059
1403
482!
1747
2091
2434
2777
3"9
3462
343
»27
xo 3804
4146
4487
5169
5510
5851
6191
6531
6871
341
128
10 7210
7549
7888
8227
8565
8903
9241
9579
9916
338
12S ]
11
0253
337
129
XI 0590
0926
1263
1599
1934
2270
2605
2940
3275
3609
335
130
XI 3943
4277
461 1
4944
5278
8595
5611
5943
6276
6608
6940
333
131
IX 7271
7603
7934
8265
8926'
9256
9586
9915
331
131 12
0245
330
132 i 12 0574
0903
4178
1231
1560
1888
2216
2544
5806
2871
3198
3525
328
m , " 3852
4504
4830
5156
8399
5481
6131
6456
6781
325
134 12 7105
7429
7753
8076
8722
9045
9368
9O90
323
1 *34 . 13
0012
323
18«
13 0334
o6«
0977
1298
1619
1939
2260
2580
2900
3219
321
'136
13 3539
3858'
4177
4496
4814
5133
5451
5769
6086
6403
318
!i37
X3 6721
7037
7354
7671
7987
8303
8618
8934
9249
9564
316
'^0
13 9879
315
138
14
0194
0508
0822
1 136
1450
1763
2076
2389
2702
3H
139
14 3o»5
3327
3639
3951
4263
4574
4885
5196
5507
S8i8
3"
140 14 6128
6438
6748
7058
7367
7676
7985
8294
8603
8911
309
141
14 9219
9527
9835
308
:Hi
^5 • •
0142
0449
0756
1063
1370
1676
1982
307
142
X5 2288
2594
2900
3205
3510
lS'5
4120
4424
4728
5032
8061
305
143
15 5336
5640
8664
5943
8965
6246
6549
6852
7154
7457
7759
303
144
15 8362
9266
9567
9868
• • • • • •
302
144
16
0168
0469
0769
1068
301
145
I 16 1368
1667
1967
2266
2564
2863
3161
3460
3758
4055
299
M46 1 16 4353
4650
4947
5244
5541
5838
6134
6430
9380
6726
7022
297
>47
i^ 7317
7613
7908 8203
8497
8792
9086
9674
9968
295
D
N)
I
2
3
4
5
6
7
8
9
40
MATHEMATICAL TABLES.
N
148
I
2
3
4
5
6
7
8
9
D
293
17 0262
0555
0848
II4I
1434
1726
2019
2311
2603
2895
149
17 3186
3478
3769
4060
4351
4641
4932
5222
S5I2
5802
291
150
17 6091
6381
6670
6959
7248
7536
7825
8113
8401
8689
289
151
17 8977
9264
9552
9839
287
151
18
0126
0413
0699
0986
1272
1558
287
IS2
z8 1844
2129
2415
2700
2985
3270
3555
3839
4123
4407
^?5
I S3
18 4691
4975
5259
5542
5825
6108
6391
6674
6956
7239
283
154
i8~ 7521
7803
8084
8366
8647
8928
9209
9490
9771
281
»54
19
0051
281
279
155
19 0332
0612
0892
II7I
145 1
1730
2010
2289
2567
2846
156
19 3125
3403
3681
3959
4237
45 »4
4792
5069
5346
5623
278
«57
19 5900
6176
6453
6729
7«>5
7281
7556
7832
8107
8382
276
'55
19 8657
8932
9206
9481
9755
275
158
2(>
0029
0303
0577
0850
1124
274
159
20 1397
1670
»943
2216
2488
2761
3033
3305
3577
3848
272
160
20> 4120
4391
4663
4934
5204
5475
8173
5746
E441
6016
6286
6556
271
161
20 6826
7096
7365
7634
7904
8710
8979
9247
269
162
20 9515
9783
268
162
21
c»5i
0319
0586
0853
1121
1388
1654
1921
267
163
21 2188
2454
2720
2986
3252
3518
3783
4049
43H
4579
266
164
21 4844
5109
5373
5638
5902
6166
6430
6694
6957
9585
7221
264
166
21 7484
7747
8010
8273
8536
8798
9060
9323
9846
262
166
22 0108
0370
0631
0892
"53
1414
1675
1936
2196
2456
261
167
22 2716
2976
3236
3496
3755
4015
4274
4533
4792
5051
259
168
22 5309
5568
5826
6084
6342
6600
6858
7115
7372
7630
258
169
22 7887
8144
5400
8657
8913
9170
9426
9682
9938
257
169
23
0193 256
170
23 0449
0704
0960
1215
1470
1724
1979
2234
2488
2742
255
171
23 2996
3250
3504
3757
401 1
4264
4517
4770
5023
5276
253
172
23 5528
23 8046
5781
S^33
6285
6537
6789
7041
7292
7544
7795
252
173
8297
8548
8799
9049
9299
9550
9800
251
173
24
• « B
0050
0300
250
174
24 0549
0799
1048
1297
1546 179s
2044
2293
2541
2790
249
175
24 3038
3286
3534
3782
4030
4277
4525
4772
5019
5266
248
176
24 55^3
5759
8219
6006
6252
6499
6745
6991
72J7
7482
7728
246
177
24 7973
8464
8709
8954
9198
9443
9687
9932
246
177
25
0176
245
178
25 0420
25 2853
0664
o9oi;
3580
1395
1638
1881
2121
4548
2368
2610
243
179
3096
3338
3822
4064
4306
4790
5031
242
180
25 5273
5514
5755
5996
6237
6477
6718
6958
7198
7439
241
181
25 7679
7918
8158
8398
8637
8877
9116
9355
9594
9833
239
182
26 0071
0310
0548
0787
1025
1263
1 501
1739
1976
2214
238
'S3
26 2451
2688
2925
3162
3399
3636
3873
4109
4346
4582
237
184
26 4818
5054
5290
5525
5761
5996
6232
6467
6702
6937
235
185
26 7172
7406
7641
7875
81 10
8344
8578
8812
9046
9279
234
186
26 9513
9746
9980
234
186
27
0213
0446
0679
0912
1144
1377
1609
233
'SZ
27 1842
2074
2306
2538
4850
2770
3001
3233
3464
3696
3927
232
188
27 4158
4389
4620
5081
53"
5542
5772
6002
6232
230
D
N
I
2
3
4
5
6
7
8
9
LOGARITHMS OF NUMBERS.
41
N 1 o
I
2
3
4
5
6
7
8
9
D
229
228
189 27 6462
6692
6921
7151
7380 7609
7838
8067
8296
8525
190 ay 8754
8982
921 1
9439
9667
9895
190 a8
a • • • «
0123
035'
0578
2849
0806
228
191 a8 1033
1261
1488
1715
1942
2169
2396
2622
3075
227
192
aa 3301
3527
3753
3979
4205
4431
4656
4882
5107
5332
226
193
a8 SSS7
i^6
6007
6232
6456
6681
6905
7130
7354
9589
7578
9812
225
194
a8 7802
8249
8473
8696
8920
9143
9366
223
195
29 0035
0257
Q480
0702
0925
1 147
1369
1591
3804
1813
2034
222
196
29 2256
2478
2699
2920
3141
3363
3584
4025
4246
221
'^Z
ag 4466
4687
4907
5127
5347
5567
5787
6007
6226
6446
220
198
39 6665
6884
7104
7323
7542
7761
7979
8198
8416
8635
219
X99
298853
9071
9289
9507
9725
9943
218
'99
30
0161
0378
0595
0813
2x8
200
30 1030
1247
1464
1681
180R
2114
233«
2547
2764
2980
217
201
30 3196
3412
3628
3844
4059
4275
4491
4706
4921
5136
216
202
30 535 >
5566
5781
5996
8137
621 1
6425
6639
6854
7068
7282
215
203
30 7496
7710
7924
8351
8564
8778
8991
9204
9417
213
204
30 9630
9843
• a • •
• « •
213
204
3X
0056
0268
0481
0693
0906
1118
1330
1542
212
1205
31 1754
1966
2177
2389
2600
2812
3023
3234
3445
3656
211
! 206
31 3867
4078
4289
4499
4710
4920
5130
5340
5551
5760
210
207
31 5970
31 8063
6180
6390
6599
6809
7018
7227
7436
7646
7854
209
208
8272
8481
8689
8898
9106
9314
9522
9730
9938
208
209
210
32 0146
0354
0562
0769
0977
1 184
1391
1598
1805
2012
207
32 2219
2426
2633
2839
3046
3252
3458
3665
3871
4077
206
211
32 4282
44^ ;8
4694
4899
5105
5310
5516
5721
5926
6131
205
212
32 6336
32 8380
6541
6745
6950
7155
7359
7563
7767
7972
8176
204
213
8583
8787
8991
9194
9398
9601
9805
204
2J3
33
0008
0211
203
2H
33 0414
0617
G819
1022
1225
1427
1630
1832
2034
2236
202
215
33 2438
2640
2842
3044
3246
3447
3649
3850
4051
4253
202
216
33 4454
4655
4856
5057
5257
5458
5658
5859
6059
6260
201
217
33 6460
6660
6860
7060
7260
7459
7659
7858
«os«
8257
200
m
218
33 8456
8656
8855
9054
9253
9451
9650
9849
200
218
34
• • •
0047
0246
199
219 34 0444
0642
0841
1039
1237
1435
1632
1830
2028
2225
198
220
34 2423
2620
2817
3014
3212
3409
3606
3802
3999
4196
197
. 221
34 4392
4589
4785
4981
5178
5374
5570
5766
5962
6157
196
222
34 6353
6549
6744
6939
7135
7330
7525
7720
7915
8110
195
223
34 8305
8500
8694
8889
9083
9278
9472
9666
9860
194
223
35
« • •
0054
1989
194
224
35 Q248
0442
c6i6
0829
1023
1216
1410
1603
1796
193
'225 35 21^3
2375
2568
2761
4876
3147
3339
3532
3724
3916
193
1 226
35 4108
4301
4493
6408
4685
5068
5260
5452
5643
5834
192
227
35 6026
6217
6599
6790
6981
7172
7363
7554
7744
191
1
228
35 7935
8125
8316
8506
86q6
8886
9076
9266
9456
9646
190
1
229
359835
189
1
1
229
35
0025
0215
0404
0593
0783
0972
1 161
1350
1539
189
D
N
I
2
3
4
5
6
7
8
9
i
42
MATHEMATICAL TABLES.
N
6
8
D
230 36 1728 191 7 2105 2294 2482
231 363612 3800 3988 4176 4363
232 36 5488 5675 5862 6049 6236
233 36 7356 7542 7729 7915 8101
234 36 9216 9401 9587 9772 9958
2671 2859 3048
4551 4739 4926
6423 6610 6796
8287 8473 8659
3236 3424
5"3 5301
6983 7169
8S45 9030
P • a •
[88
[88
187
[86
[86
234
235
236
237
238
239
239
37
37 1068
37 2912
37 4748
37 6577
37 8398
38
1253 1437 1622 1806
3096 3280 3464 3647
4932 5"5 5298 5481
6759 6942 7124 7306
8580 8761 8943 9124
0143
1991
3831
5664
7488
9306
0328
2175
4015
5846
7670
9487
05 '3
2360
4198
6029
7852
9668
0698
2544
4382
6212
8034
9849
0883
2728
4565
6394
8216
0030
(85
[84
[84
^83
[S2
[82
[81
240
241
242
243
244
38 021 I
38 2017
38 3815
38 5606
38 7390
0392
2197
3995
578§
7568
0573
2377
4174
5964
7746
0754
2557
4353
6142
7923
0934
2737
4533
6321
8101
I I 15 1296 1476
2917 3097 3277
4712 4891 5070
6499 6677 6856
8279 8456 8634
1656 1837 181
3456 3636 180
5249 5428 179
7034 7212 178
8S11 8989 178
245
245
246
247
248
249
38 9166
39
39 0935
39 2697
39 4452
39 6199
9343 9520 9698 9875
1112
2873
4627
6374
1288
3048
4802
6548
1464
3224
4977
6722
1641
3400
5152
6896
0051
1817
3575
5326
7071
0228
1993
3751
5501
724s
0405
2169
3926
5676
7419
0582
2345
4101
5850
7592
0759
2521
4277
6025
7766
77
77
76
76
75
74
250
251
251
252
253
254
255
256
257
257
258
259
39 7940
39 9674
40
40 1401
40 3121
40 4834
8114
9847
1573
3292
5cx>s
8287 8461 8634
0020
1745
3464
S176
0192
1917
3635
5346
0365
2089
3807
5517
8808 8981 9154 9328 9501
0538
2261
3978
$688
071 1
2433
4149
5858
0883
2605
4320
6029
1056
2777
4492
6199
1228
2949
4663
6370
73
73
73
72
71
71
40 6540
40 8240
40 9933
41
41 1620
41 3300
6710
8410
0102
1788
3467
6881
8579
0271
1956
3635
7051
8749
• • • • •
0440
2124
3803
7221
8918
0609
2293
3970
739X
9087
0777
2461
4137
7561
9257
• •
0946
2629
4305
7731
9426
1114
2796
4472
7901
9595
1283
2964
4639
8070
9764
1451
3>32
4806
70
[69
[69
[69
[68
67
260
261
262
263
263
264
41 4973
41 6641
41 8301
41 9956
42
42 1604
5140
6807
8467
0121
1768
5307
6973
8633
0286
1933
5474
7139
8798
0451
2097
5641 ] 5808
7306
8964
0616
2261
7472
9129
0781
2426
5974
7638
9295
0945
2590
6141
7804
9460
1 110
2754
6308
7970
9625
1275
2915
6474
8135
9791
1439
3082
[67
[66
[6s
[65
[65
[64
265
266
267
268
269
269
270
271
N
42 3246
42 4882
42 651 I
42 8135
42 9752
43
3410
5045
6674
8297
9914
3574
5208
6836
8459
3737
5371
6999
8621
3901
5534
7161
8783
0075 0236 0398
4065
5697
7324
8944
4228
5860
7486
9106
4392
6023
7648
9268
4555
6186
781 1
9429
4718
6349
7973
9591
0559 0720 0881 1042 1203
43 1364
43 2969
1525
3130
1685
3290
1846
3450
2007
3610
2167
3770
2328
3930
2488
4090
7
2649
4249
8
2809
4409
[64
163
t62
[62
[62
[61
[61
60
D
LOGARITHMS OF NUMBERS.
43
N
I
2
3
4
5
6
7
8
9
D
272
43 45^
4729
4888
5048
5207
5367
5526
5685
5844
6004
VS9
273
43 6163
6322
64S1
6640
6799
6957
7116
7275
7433
7592
159
274
43 7751
7909
8067
8226
8384
8542
8701
8859
9017
9175
158
275
43 9333
9491
9648
9806
9964
158
275
44
0122
0279
0437
0594
0752
158
276
44 09C39
1066
1224
1381
iS3^
1695
1852
2009
2166
2323
157
277
44 2480
2637
2793
2950
3106
3263
3419
3576
3732
3889
157
278
44 4045
4201
4357
4513
4669
4825
4981
5137
5293
5449
156
279
44 5604
5760
5915
6071
6226
6382
6537
6692
6848
7003
155
280
44 7158
Z3'3
7468
7623
7778
7933
8088
8242
8397
8552
155
281
44 8706
8S61
9015
9170
9324
9478
9633
9787
9941
154
281
45
0095
154
282
45 0249
0403
0557
071 1
0865
1018
1 172
1326
1479
1633
154
^S^
45 1786
1940
2093
2247
2400
2553
2706
2859
3012
3165
153
284
45 3318
3471
3624
3777
3930
4082
4235
4387
4540
4692
153
285
45 4845
4997
5150
5302
5454
5606
5758
5910
6062
6214
^52
286
45 6366
6518
6670
6821
6973
7125
7276
7428
7579
7731
152
287
45 7882
«033
8184
8336
8487
8638
8789
8940
9091
9242
151
288
45 9392
9543
9694
9845
9995
151
2SS
46 ......
0146
0296
0447
0597
0748
151
289
46 0S98
1048
1198
1348
1499
1649
1799
1948
2098
2248
150
280
46 2398
2548
2697
2847
2997
3146
3296
3445
3594
3744
150
291
46 3893
4042
4191
4340
4490
4639
4788
4936
5085
5234
149
292
46 5383
46 6868
5532
5680
5829
5977
6126
6274
6423
6571
6719
149
293
7016
7164
7312
7460
7608
7756
7904
8052
8200
148
294
46 8347
8495
9969
8643
8790
8938
9085
9233
9380
9527
9675
148
295
45 9822
• B •
147
295
47
01 16
0263
0410
0557
0704
0851
0998
1 145
147
296
47 1292
1438
I5«5
1732
1878
2025
2171
2318
2464
2610
146
297
47 2756
2903
3049
3195
3341
3487
3633
3779
3925
4071
146
298
47 4216
4362
4S08
4653
4799
4944
5090
5235
5381
5526
146
299
47 5671
5816
5962
6107
6252
6397
6542
6687
6832
6976
145
300
47 7121
7266
741 1
7555
7700
7844
7989
8133
8278
8422
145
301
47 8566
871 1
8855
8999
9143
9287
9431
9575
9719
9863
144
3«>2
48— 0007
0151
1586
0294
0438
0582
0725
0869
1012
"§S
1299
144
393
48 1443
1729
1872
2016
2K9
2302
2445
2588
2731
143
304
48 2874
3016
3^59
3302
3445
3587
3730
3872
4015
4157
M3
305
48 4300
4442
4585
4727
4869
5011
5153
5295
5437
5579
142
306
43 5721
5863
6005
6147
6289
6430
6572
6714
6855
6997
142
3^
48 7138
7280
7421
7563
7704
7845
7986
8127
8269
8410
141
SoS
48 8551
8692
8833
8974
9114
9255
9396
9537
9677
9818
141
3P9
48 9958
140
. 3P9
49
0099
0239
0380
0520
0661
0801
0941
108 1
1222
140
310
49 1362
1502
1642
1782
1922
2062
2201
2341
2481
2621
140
3^1
49 2760
29CX>
3040
3179
3319
3458
3597
3737
3876
4015
139
312
49 4155
4294
4433
4572
471 1
4850
4989
5128
5267
5406
139
313
49 5544
5683
5822
5960
6099
6238
6376
6515
6653
6791
139
' 3«4
49 6930
7068
7206
7344
7483
7621
7759
7897
8035
8173
138
N
I
2
3
4
5
6
7
8
9
D
44
MATHEMATICAL TABLES.
N
315
316
316
317
318
319
49 83"
49 9687
50 1059
50 2427
50 3791
8448
9824
1 196
2564
3927
8586
9962
1333
2700
4063
8724 8862
• •• •
0099
1470
2837
4199
8
8999 9137 9275 9412 9550
0236 0374
1607 1744
2973
4335
3109
4471
0511
1880
3246
4607
0648
2017
3382
4743
0785
2154
35^8
4878
0922
2291
36SS
5014
D
138
^37
t37
[37
^36
^36
320
321
322
323
323
324
50 5150
50 6505
50 7856
50 9203
51
51 0545
5286
6640
7991
9337
5421
6776
8126
9471
5557
691 1
8260
9606
5693
7046
8395
9740
0679 0813 0947 1081
5828
7181
8530
9874
1215
5964
7316
8664
0009
1349
6099
7451
8799
0143
1482
6234
7586
8934
0277
1616
6370
7721
9068
041 1
1750
^36
135
^35
34
'34
134
325
326
327
328
329 '
51 i88j
51 3218
51 4548
51 5874
51 7196
2017
3351
4681
6006
7328
2151
3484
4813
6139
7460
2284
3617
4946
6271
7592
2418
3750
5079
6403
7724
2551
3883
521 1
6535
785s
2684
4016
5344
6668
7987
2818
4149
5476
6800
81 19
2951
4282
5609
6932
8251
3084
4415
5741
7064
8382
^33
'33
133
[32
'32
380
331
331
332
333
334
51 8514
51 9828
52
52 1138
52 2444
52 3746
8646
9959
1269
2575
3876
8777 8909 9040
0090
1400
2705
4006
022 X
IJ30
2835
4136
0353
1661
2966
4266
9171 9303 9434 9566 9697
0484
1792
3096
4396
0615
1922
3226
4526
0745
2053
3356
4656
0876
2183
3486
4785
1007
2314
3616
4915
'31
31
'31
31
[30
'30
385
336
337
338
338
339
52 5045
52 6339
52 7630
52 8917
53
53 0200
5174
6469
7759
9045
5304
6598
7888
9174
5434
6727
8016
9302
5563
6856
8145
9430
5693
6985
8274
9559
5822
7114
8402
9687
5951
7243
8531
9815
6081
7372
8660
9943
0328 0456 0584 0712
0840 0968 1096 1223
6210
7501
8788
0072
1351
[29
[29
129
[28
[28
28
840
341
342
343
344
53 1479
53 2754
53 4026
53 5294
53 6558
1607
2882
4153
5421
6685
1734
3009
4280
5547
6811
1862
3136
4407
5674
6937
1990
3264
4534
5800
7063
2117
3391
4661
5927
7189
2245
3518
4787
6053
7315
2372
3645
4914
6180
7441
2500
3772
5041
6306
7567
2627
3899
5167
6432
7693
[28
[27
[27
[26
[26
346
346
346
347
348
349
53 7819
53 9076
54
54 0329
54 1579
54 2825
7945
9202
0455
1704
2950
8071
9327
0580
1829
3074
8197
9452
0705
1953
3199
8322
9578
0830
2078
3323
8448
9703
• • • •
0955
2203
3447
8574
9829
1080
2327
3571
8699
9954
1205
2452
3696
8825 8951
0079
1330
2^76
3820
0204
1454
2701
3944
[26
[26
[25
[25
25
124
360
351
352
353
354
354
365
356
357
N
54 4068
54 5307
54 6543
54 7775
54 9003
55
4192
5431
6666
7898
9126
4316
5555
6789
8021
9249
4440
5678
6913
8144
9371
4564
5802
7036
8267
9494
4688
5925
7159
8389
9616
4812
6049
7282
8512
9739
4936
6172
7405
863s
9861
5060
6296
7529
8758
9984
5183
6419
7652
8881
0106
55 0228 0351 0473 0595 0717
55 '450 1572" 1694 1816 1938
55 2668 2790 29 I I 3033 3155
0840
2060
3276
0962
2181
3398
1084
2303
3519
1206
2425
3640
8
1328
2547
3762
[24
124
'23
23
23
'23
[22
[22
[21
D
LOGARITHMS OF NUMBERS.
45
N
I
2
3
4
5
6
7
8 9
D
121
358
55 3883 4004
4126
4247
4368
4489
4610
4731
4852 4973
359
55 5094 52 » 5
5336
5457
5578
5699
5820
5940
6061 6182
121
360 55 6303 6423
6544
6664
6785
6905
8108
7026
7146
7267 7387
120
361 55 7507 7627
7748
7868
7988
8228
8349
8469 8589
120
362 1 55 8709 8829
8948
go68
91^
9308
9428
9548
9667 9787
120
363 55 9907
120
363
56 0026
0146
0265
0385
0504
1698
0624
0743
0863 0982
"9
364
56 iioi 1221
1340
H59
1578
1817
1936
2055 2174
119
365 56 2293 2412
2531
2650
2769
2887
3006
3125
3244 3362
119
366 56 3481 3600
37»8
3837
3955
4074
4192
43"
4429 4548
119
367
56 4666 4784
4903
5021
5139
5257
5376
5494
5612 5730
118
368
56 5848 5966
6084
6202
6320
6437
6555
6673
6791 6909
118
369
56 7026 7144
7262
7379
7497
7614
7732
7849
7967 8084
118
370 56 8202 8319
8436
8554
8671
8788
8905
9023
9140 9257
117
371
55 9374 9491
9608
9725
9842
9959
^
117
37 «
57 ]•..
0076
0193
0309 0426
"7
372
57 0543 0660
0776
0893
2058
lOIO
1 126
1243
1359
1476 1592
117
373
57 1709 1825
1942
2174
2291
2407
2523
2639 2755
116
374 57 2872 2988
3»04
3220
3336
3452
3568
3684
3800 3915
116
375 57 403' 4147
4263
4379
4494
4610
4726
4841
4957 5072
116
376
57 5188 5303
S419
^^
5650
5765
5880
5996
61 1 1 6226
"5
377
57 6341 6457
6572
6802
6917
7032
7147
8295
7262 7377
"5
378
57 7492 7607
7722
7836
7951
8066
81S1
8410 8525
"5
379
57 8639 8754
8868
8983
9097
9212
9326
9441
9555 9669
114
380 57 9784 9898
■• «■■ •«■••■
"4
380 58
0012
0126
0241
0355
0469
0583
0697 081 I
114
381
58 092s 1039
"53
1267
1381
1495
1608
1722
1836 1950
2972 3085
"4
382
58 2063 2177
2291
2404
2518
2631
2745
2858
114
383
58 3199 33»2
3426
3539
3652
3Z^§
3879
3992
4105 4218
"3
^
58 4331 444*
4557
4670
4783
4896
5009
5122
5235 5348
"3
385
58 5461 5574
5686
5799
5912
6024
6137
6250
6362 6475
"3
386
58 6587 6700
6812
6925
7037
7149
P^^
7374
7486 7599
112
387
58 77" 7823
7935
8047
8160
8272
8384
8496
8608 8720
112
38S
58 8832 8944
9056
9167
9279
9391
9503
9615
9726 9838
112
. 389
58 9950
112
389
59 0061
0173
0284
0396
0507
0619
0730
0842 0953
112
300
59 1065 1 1 76
1287
1399
15 10
I62I
1732
1843
1955 2066
III
391
59 2177 2288
2399
2510
2621
2732
2843
2954
3064 3175
III
392
59 3286 3397
3«>o8
3618
3729
3840
3950
4061
41 7 I 4282
III
393
59 4393 4503
4614
4724
4834
4945
5055
5165
5276 5386
110
394
59 5496 . 5606
5717
5827
5937
6047
6157
6267
6377 6487
no
395
59 6597 6707
6817
6927
7037
7146
7256
7366
7476 7586
no
396
59 7695 7805
59 879* 8900
7914
8024
8134
8243
8353
8462
8572 8681
no
397
9009
9119
9228
9337
9446
9556
9665 9774
109
398
59 9883 9992
109
398
60
OIOI
0210
0319 I 0428
0537
0646
0755 ^864
109
399
400
60 0973 1082
II9I
1299
1408 ^ 1517
1625
1734
1843 1951
109
60 2060 2169
2277
2386
2494
2603
5
271 1
6
2819
7
2928 3036
8 9
108
N
1
I
2
3
4
46
MATHEMATICAL TABLES.
N
I
2
3
4
5
6
7
8
9
D
108
401
60 3144
3253
3361
3469
3577
3686
3794
3902
4010
41 18
402
60 4226
4334
4442
^§50
4658
4766
4874
4982
5089
5197
108
403
60 5305
5413
5521
5628
5736
5844
595'
6059
6166
6274
108
404
60 6381
6489
6596
6704
6811
6919
7026
7133
7241
7348
107
405
60 7455
7562
7669
SJ?
7884
\ 7991
8098
8205
8312
8419
107
406
60 8526
8633
8740
8954 ; 9061
9167
9274
9381
9488
107
407
60 9594
9701
9808
9914
107
407
61
0021
0128
0234
0341
0447
0554
107
408
6x 0660
0767
0^73
0979
1086
1 192
1298
1405
1511
1617
106
409
61 1723
1829
1936
2042
2148
2254
2360
2466
2572
2678
X06
410
61 2784
2890
2996
3102
3207
3313
3419
3525
3630
4686
3736
106
411
61 3842
3947
4053
4159
4264
4370
4475
4581
4792
106
412
61  4897
5003
5108
5213
5319
5424
5529
5634
5740
5845
105
413
61 5950
6055
6160
6265
6370
6476
6581
6686
6790
6895
105
414
61 7000
7105
7210
7315
7420
7525
7629
7734
7839
7943
105
415
61 8048
8153
9198
8257
8362
8466
8571
8676
8780
8884
8989
105
416
619093
9302
9406
9511
9615
9719
9824
9928
105
4x6
6a
«■••••
0032
104
417
62 0136
0240
1280
0344
0448
0552
0656
0760
0864
0968
1072
104
418
62 1 1 76
1384
1488
1592
1695
1799
1903
2007
21 10
104
419
62 2214
2318
2421
2525
2628
2732
2835
2939
3042
3146
104
420
62 3249
3353
4488
3559
3663
3766
3869
3973
4076
4179
103
421
62 4282
4385
4591
4695
4798
4901
5004
5107
5210
103
422
6a 5312
5415
5518
5621
5724
5827
5929
6032
6135
6238
»03
423
62 6340
6443
6546
6648
6751
^^53
6956
7980
Z°5^
7161
P^^
103
424
62 7366
7468
7571
7673
7775
7878
8082
8185
8287
102
425
62 8389
8491
8593
8695
8797
8900
9002
9104
9206
9308
102
426
62 9410
9512
9613
9715
9817
9919
102
426
63
0021
0123
0224
0326
102
427
63 0428
0530
0631
0733
^35
0936
1038
1 139
1241
1342
102
428
63 1444
1545
1647
1748
1849
1951
2052
2153
2255
2356
lOI
429
63 2457
2559
2660
2761
2862
2963
3064
3«65
3266
3367
lOI
480
63 3468
3569
3670
3771
3872
3973
4074
4175
4276
4376
lOI
431
63 4477
4578
4679
4779
4880
4981
5081
5182
5283
5383
lOI
432
63 5484
5584
5685
5785
5886
5986
6087
6ih7
6287
6388
100
433
63 6488
6588
6688
6789
6889
6989
7089
7189
7290
7390
8389
100
434
63 7490
7590
7690
7790
7890
7990
8090
8190
8290
100
485
63 8489
8589
8689
8789
8888
8988
9088
9188
9287
9387
100
436
63 9486
9586
9686
9785
9885
9984
lOO
436
64
0084
0183
0283
0382
99
437
64 0481
0581
0680
0779
0879
0978
1077
"77
1276
1375
99
438
64 1474
1573
1672
1771
1871
1970
2069
2168
2267
2366
99
439
64 2465
2563
2662
2761
2860
2959
3946
3058
4044
3156
4143
3255
4242
3354
4340
99
99
440
64 3453
3551
3650
3749
3847
441
64 4439
4537
4636
4734
4832
4931
5029
5127
5226
5324
98
442
64 5422
5521
5619
5717
5815
5913
601 1
6110
6208
6306
98
443
64 6404
6502
6600
6698
6796
6894
6992
7089
7187
7285
98
444
N
64 7383
7481
7579
7676
7774
7872
7969
8067
8165
8262
98
I
2
3
4
s
6
7
8
9
D
LOGARITHMS OF NUMBERS.
47
N o
P
I
2
3
4
S
6
7
8
9
D
445
64 8360
8458
8555
8653
8750
8848
8945
9043
9140
9237
97
446
64 9335
9432
9530
9627
9724
9S21
9919
97
446
65
0016
0113
0210
97
447
65 0308
0405
0502
0599
0696
0793
0890
0987
1084
1181
97
i 44S
65 1278
1375
1472
1569
1666
1762
'S^?
1956
2053
2150
97
449
65 2246
2343
2440
2536
2633
2730
2826
2923
3019
311^
97
450
65 3213
3309
3405
3502
3598
3695
3791
3888
3984
4080
96
4S»
65 4177
4273
4369
4465
4562
4658
4754
4850
4946
5042
96
452
65 5138
5235
5331
5427
5523
5619
5715
5810
5906
6002
96
453
65 6098
6194
6290
6386
6482
6577
6673
6769
6864
6960
96
454 65 7056
7152
7247
7343
7438
7534
7629
7725
7820
7916
96
455 65 Soil
8107
8202
8298
8393
8488
8584
8679
8774
8870
95
456
65 8965
9060
9155
9250
9346
9441
9536
963^
9726
9821
95
457
65 9916
••■•»■
95
457
66
OOII
0106
0201
0296
0391
0486
0581
0676
0771
95
45^
66 0865
0960
1055
1 150
1245
1339
1434
2380
1529
1623
1718
95
459
66 1813
1907
2002
2096
2191
2286
2475
2569
2663
95
460
65 2758
2852
2947
3041
3135
3230
3324
3418
3512
3607
94
461
66 3701
3795
3889
3983
4078
4172
4266
4360
4454
4548
94
462
66 4642
4736
4830
4924
5018
5112
5206
5299
5393
5487
94
463
66 5581
5675
5769
5862
5956
6050
6143
6237
6331
6424
94
464
66 6518
6612
6705
6799
6892
6986
7079
7173
7266
7360
94
465
66 7453
7546
7640
7733
7826
7920
8013
8106
8199
8293
93
466
66 8386
8479
8572
8665
8759
88^2
8945
9038
9J31
9224
93
467
66 9317
9410
9503
9596
9689
9782
9875
9967
93
467
67
• • • a •
0060
0153
1080
93
46S
67 0246
0339
0431
0524
0617
0710
0802
089s
0988
93
469
67 "73
1265
1358
145 I
1543
1636
1728
1821
1913
2005
93
470
67 2098
2190
2283
2375
2467
2560
2652
2744
2836
2929
92
471
67 3021
3"3
3205
3297
3390
3482
3574
3666
3758
3850
92
472
67 3942
4034
4126
4218
4310
4402
4494
4586
4677
4769
92
473
67 4861
4953
5045
5137
5228
5320
5412
5503
5595
5687
92
474
67 5778
5870
5962
6053
6145
6236
6328
6419
6511
6602
92
475 67 6694
6785
7698
6876
6968
7059
7151
7242
8154
7333
7424
8336
7516
91
476
67 7607
67 8518
7789
7881
7972
8063
8245
8427
91
477
8609
8700
8791
8882
^Z3
9064
9155
9246
9337
91
478
67 9428
9519
9610
9700
9791
9882
9973
91
478
68
C063
0154
0245
91
479
68 0336
0426
0517
0607
0698
0789
0879
0970
1060
1151
91
480
68 1241
1332
1422
I5I3
2416
1603
1693
1784
1874
1964
2055
90
481
68 2145
2235
2326
2506
2596
2686
2777
2867
2957
90
4S2
68 3047
3137
3227
3317
3407
3497
3557
3^77
3767
3857
90
: 483 68 3947
4037
4127
4217
4307
4396
4486
4576
4666
4756
90
484
68 4845
4935
5025
5II4
5204
5294
5383
5473
5563
5652
90
485
68 5742
5831
5921
6010
61CX)
6189
6279
6368
6458
6547
89
486
68 6636
6726
6815
6904
6994
7083
7172
7261
7351
7440
89
487
68 7529
7618
7707
7796
7886
7975
8064
8153
8242
8331
89
488
68 S420
8509
8598
8687
8776
8865
8953
9042
9131
9220
89
1489
68 9309
9398
9486
9575
9664
9753
5
9841
9930
89
D
j N }
I
2
3
4
6
7
8
9
48
MATHEMATICAL TABLES.
N
4«9
1490
491
492
493
494
69
8
0019 0107
69 0196
69 1081
69 1965
69 2847
693727
0285
1 170
2053
2935
381S
0373
1258
2142
3023
3903
0462
1347
2230
3111
3991
0550
1435
2318
3>99
4078
0639
1524
2406
3287
4166
0728
1612
2494
3375
4254
0816
1700
2583
3463
4342
0905
17S9
2671
3551
4430
0993
1877
2759
3639
4517
D
89
89
88
88
88
88
496
496
497
498
499
69 4605
69 5482
69 6356
69 7229
69 8101
4693
5569
6444
7317
8188
4781
5657
6531
7404
8275
4868
5744
6618
7491
8362
4956
5832
6706
7578
8449
5044
5919
6793
7665
8535
513"
6007
6880
7752
8622
5219
6094
6968
7839
8709
5307
6182
7055
7926
8796
5394
6269
7142
8014
8883
88
87
87
87
87
500
501
502
503
504
69 8970
69 9838
70
70 0704
70 1568
70 2431
9057
9924
0790
1654
2517
9144 9231 9317
001 1
0877
1741
2603
0098
0963
1827
2689
0184
1650
1913
2775
9404 9491 9578 9664 9751
0271
1 136
1999
2861
0358
1222
2086
2947
0444
1309
2172
3033
0531
1395
2258
3"9
0617
1482
2344
3205
87
87
87
86
86
86
505
506
507
508
509
510
511
512
512
5*3
514
70 3291
70 4151
70 5008
70 5864
70 6718
3377
4236
5094
5949
6803
3463
4322
5179
60
6^
3549
4408
5265
6120
6974
3635
4494
5350
6206
7059
3721
4579
5436
6291
7144
3807
4665
5522
6376
7229
3893
4751
5607
6462
7315
3979
4837
5693
6547
7400
4065
4922
5778
6632
7485
70 7570
70 8421
70 9270
71
71 0117
71 0963
7655
8506
9355
0202
1048
7740
8591
9440
0287
1 132
7826
8676
9524
0371
1217
791 1
8761
9609
0456
1301
7996
8846
9694
0540
1385
8081
8931
9779
0625
1470
8166
9015
9S63
0710
1554
8251
9100
9948
0794
1639
8356
9185
• • • • •
0033
0879
1723
86
86
86
85
85
f5
85
85
84
516
516
518
5«9
71 1807
71 2650
71 3491
71 4330
71 5*67
1892
2734
3575
4414
5251
1976
2818
3659
4497
5335
2060
2902
3742
4581
5418
2144
2986
3826
4665
5502
2229
3070
3910
4749
5586
2313
3>54
3994
4833
5669
2397
3238
4078
4916
5753
2481
3323
4162
5000
5836
2566
3407
4246
5084
5920
84
84
84
84
84
520
, 521
522
i 523
524
524
71 6003
71 6838
71 7671
71 8502
71 9331
72
6087
6921
7754
8585
9414
6170
7004
7837
8668
9497
6254
7088
7920
8751
9580
6337
7171
8003
8834
9663
6421
7254
8086
8917
9745
6s(H
7338
8169
9000
9828
6588
7421
8253
9083
9911
6671
7504
8336
9165
9994
6754
7587
8419
9248
0077
7a 01 g9 0242 0325 0407 0490
72 0986 1068 I 151 1233 1316
7a i8u 1893 1975 2058 2140
7a 2634 2716 2798 2881 2963
72 3456 3538 3620 3702 3784
0573 0655 0738 0821 0903
1398 1481 1563 1646 1728
2222 2305 2387 2469 2552
3045 3127 3209 3291 3374
3866 3948 4030 41 12 4194
72 4276
4358
4440
4522
4604
72 5095
5176
5258
5340
5422
72 5912
5993
6075
61 s6
6238
7a 6727
0^
6890
6972
7053
4685 4767 4849 4931 50'3
5503 5585 5667 5748 5830
6320 6401 64S3 6564 6646
7134 7216 7297 7379 7460
«3
83
P
83
83
82
82
82
82
82
82
82
81
8
D
LOGARITHMS OF NUMBERS.
49
1
1
1«
N
534
I
2
3
4
5
6
7
8
9
D
81
81
72 7541
7623
7704
7785
7866
7948
8029
8110
8191
8273
9084
»d5 1
7^ 8354
8435
8516
8597
8678
8759
8841
8922
9003
536
72 9165
9246
9327
9408
9489
9570
9651
9732
9813
9893
81
537
72 9974
^
81
537 73
OOS5
0136
0217
0298
0378
0459
0540
0621
0702
81
1 538 1
73 0782
086^
0944
1024
1105
1186
1266
1347
1428
1508
81
.{
539
73 1589
1669
1750
1830
1911
1991
2072
2152
2233
2313
81
HO
73 2394
2474
2S55
2635
3438
2715
2796
2876
2956
3037
3"7
80
541
73 3>97
3278
3358
3518
3598
3679
3759
3839
3919
80
542
73 3999
4079
4160
4240
4320
4400
4480
4560
4640
4720
80
543
73 4800
4880
4960
5040
5120
5200
5279
5359
5439
55>9
80
I
544
73 5599
5679
5759
5838
5918
5998
6078
6157
6237
6317
io
546
73 6397
6476
6556
6635
6715
6795
6874
6954
7034
7113
80
546
73 7193
7272
7352
7431
751 1
7590
7670
7749
7829
7908
79
547
73 7987
8067
8x46
8225
8305
8384
8463
8543
8622
8701
79
548
73 8781
8860
8939
9018
9^7
9'?Z
9256
9335
9414
9493
79
549
73 9572
9651
9731
9810
9889
9968
79
549
74
0047
0126
0205
0284
79
i
650
74 0363
0442
0521
0600
0678
0757
0836
0915
0994
'2?3
79
.'551
74 1152
1230
1309
1388
1467
1546
1624
1703
1782
i860
79
552
74 1939
2018
2096
2175
2254
2332
241 1
2489
2568
2647
79
553 74 2725
2804
2882
2961
3039
3II8
3196
3275
3353
3431
78
554
74 35>o
3588
3667
3745
3823
3902
3980
4058
4136
4215
78
n
555 1 74 4293
4371
4449
4528
4606
4684
4762
4840
4919
4997
75
556 74 5075
5153
5231
5309
5387
S465
5543
5621
5699
5777
78
557
74 5855
5933
601 1
6089
6167
6245
6323
6401
6479
6556
78
558
74 6634
6712
6790
6868
6945
7023
7101
7179
7256
7334
78
559
74 7412
7489
7567
7645
7722
7800
7878
7955
8033
8110
78
560
74 81S8
8266
8343
8421
8498
8576
8653
8731
8808
8885
77
;50i
748963
9040
9118
9195
9272
9350
9427
9504
9582
9659
77
562 i 74 9736
Q814
9891
77
562 ; 75
0045
0123
0200
0277
0354
0431
77 N
563
75 0508
0586
ii663
0740
0817
0894
0971
1048
1 125
1202
77
■564
75 1279
1356
1433
1510
1587
1664
1 741
1818
1895
1972
77
566
75 2048
2125
2202
2279
2356
2433
2509
2586
2663
2740
77
>66
75 2816
2893
2970
3047
3123
3200
3277
3353
3430
3506
77
'^
75 3583
S66o
3736
3813
3889
3966
4042
41 19
4195
4272
77
568
75 4348
4425
4501
4578
4654
4730
4807
4883
4960
5036
76
S^
75 5"2
5189
5265
5341
5417
5494
5570
5646
5722
5799
76
570
75 5875
595 «
6027
6103
6180
6256
6332
6408
6484
6560
76
571
75 6636
6712
6788
6864
6940
7016
7092
7168
7244
7320
227?
76
572
75 7396
7472
7548
7624
7700
7775
7851
7927
8003
76
573 ' 75 8155
8230
8988
8306
8382
8458
8533
8609
8685
8761
8836
76
574 , 75 8912
9063
9139
9214
9290
9366
9441
9517
9592
76
575
75 9668
9743
9819
9894
9970
76
575
75
• » • •
0045
0121
0196
0272
0347
75
576
75 0422
0498
0573
0649
0724
0799
0875
0950
1025
IIOI
75
577
N
76 1176
1251
1326
1402
1477
1552
1627
1702
1778
1853
75
D
I
2
3
4
5
6
7
8
9
so
MATHEMATICAL TABLES.
N
578
I
2
3
4
5
6
7
8
9
D
75
76 1928
2003
2078
2153
2228
2303
2378
2453
2529
2604
579
76 2679
2754
2829
2904
2978
3053
•
3128
3203
3278
3353
75
580
76 3428
3503
3578
3653
3727
3802
3877
3952
4027
4101
75
5^'
76 4176
4251
4326
4400
4475
4550
4624
4699
4774
4848
75
5^^
76 4923
4998
5072
5147
5221
5296
5370
5445
5520
5594
75
553
76 5669
5743
5818
5892
5966
6041
6115
6190
6264
6338
7082
74
584
76 6413
6487
6562
6636
6710
6785
6859
6933
7007
74
74
585
76 7156
7230
7304
7379
7453
7527
7601
7675
7749
7823
586
76 7898
7972
' 8046
8120
8194
8268
8342
8416
8490
8564
74
5^Z
76 8638
8712
8786
8860
8934
9008
9082
9156
9230
9303
74
588
76 9377
9451
9525
9599
9673
9746
9820
9894
9968
74
588
77
••••••
0042
74
589
77 0115
0189
0263
0336
0410
0484
0557
0631
0705
0778
74
590
77 0852
0926
0999
IP73
1 146
1220
129^
1367
1440
1514
74
591
77 1587
1661
1734
180B
1881
^9?S
2028
2102
2175
2248
73
592
77 2322
2395
2468
2542
2615
2688
2762
2835
2908
2981
73
593
77 3055
3128
3201
3274
3348
3421
3494
3567
4298
3640
3713
73
594
77 3786
3860
3933
4006
4079
4152
4225
4371
4444
73
595
77 4517
4590
4663
4736
4809
48S2
4955
5028
5100
5173
73
596
77 5246
5319
5392
5465
5538
5610
5683
5756
5829
5902
73
597
77 5974
6047
6120
6193
6265
6338
641 1
6483
6556
6629
73
598
77 6701
6774
6846
6919
6992
7064
7137
7209
7282
7354
73
599
77 7427
7499
7572
7644
77'7
7789
7862
7934
8006
8079
72
600
77 8151
i^224
8296
8368
8441
8513
8585
8658
9380
8730
8802
72
601
77 8874
8947
9019
9091
9163
9236
9308
9452
9524
72
602
77 9596
9669
9741
9813
9885
9957
• • • • « •
72
602
78
0029
OIOI
0173
0245
72
603
78 0317
0389
0461
0533
0605
0677
0749
0821
0893
0965
72
604
78 1037
IIO9
1181
1253
1324
1396
1468
1540
1612
1684
72
605
78 1755
1827
1899
1971
2042
2114
2186
2258
2329
2401
72
606
78 2473
2544
2616
2688
2759
2831
2902
2974
3046
3117
72
607
78 3189
3260
3332
340.S
3475
3546
3618
3689
3761
3832
71
608
78 3904
3975
4046
41 18
4189
4261
4332
4403
4475
4546
71
609
78 4617
4689
4760
4831
4902
4974
5045
5116
5187
5259
71
610
78 5330
5401
5472
5543
5615
5686
5757
5828
5899
5970
71
611
78 6041
6112
6183
6254
6325
6396
6467
6538
6609
6680
71
612
78 6751
6822
6893
6964
7035
7106
7177
7248
7319
7390
71
613
78 7460
753>
7602
7673
7744
7815
7885
7956
8027
8098
71
614
78 8168
8239
8310
8381
84s I
8522
8593
8663
8734
8804
71
615
78 8875
8946
9016
9087
9157
9228
9299
9369
9440
9510
71
616
78 9581
9651
9722
9792
9863
9933
70
tfi6
79
cxx}4
0074
0144
0215
70
617
79 0285
0356
0426
0496
0567
0637
0707
0778
0848
0918
70
618
79 0988
1059
1 129
1 199
1269
1340
1410
1480
1550
1620
70
619
620
79 I 69 I
1761
1831
1901
1971
2041
2111
2181
2252
2322
70
79 2392
2462
2532
2602
2672
2742
2812
28S2
2952
3022
70
621
79 3092
3162
3231
3301
3371
3441
35"
3581
3651
3721
70
622
N
79 3790
3860
3930
4000
4070
4139
4209
4279
4349
4418
70
D
I
2
3
4
_ 5
6
7
8
9
LOGARITHMS OF NUMBERS.
SI
Nj
I
«
2
3
4
5
6
7
'8
9
D
70
623
79 4488
4558
4627
4697
4767
4836
4906
4976
5045
5"S
624 79 5185
5254
5324
5393
5463
5532
5602
5672
5741
5811
70
625 79 5880
5949
6019
6088
6158
6227
6297
6366
6436
6505
69
626 79 6574
6644
6713
6782
6852
6921
6990
7060
7129
7198
69
627
79 7268
7337
7406
7475
7545
7614
7683
7752
7821
7890
69
628
79 7960
8029
8098
8167
8236
8305
8374
8443
8513
8582
69
629
798651
8720
8789
8858
8927
8996
9065
9134
9203
9272
69
630
631
79 9341
9409
9478
9547
9616
9685
9754
9823
9892
9961
69
80 0029
0098
0167
0236
0305
0373
0442
0511
0580
0648
69
632
80 0717
0786
o8s4
0923
0992
1061
1 129
X198
1266
1335
69
633
80 1404
1472
1541
1609
1678
1747
1815
1884
1952
2021
69
634
80 2089
2158
2226
2295
2363
2432
2500
2568
2637
2705
69
635
802774
2842
2910
2979
3047
3116
3184
3252
3321
3389
68
636
803457
3525
3594
3662
3730
3798
3867
3935
4003
4071
68
637
80 4139
4276
4344
4412
4480
4548
4616
4685
4753
68
638
8a 4821
4889
4957
5025
5093
5161
5229
5297
5365
5433
68
639
80 5501
5569
5637
5705
5773
5841
5908
5976
6044
6112
68
640
80 6180
6248
6316
6384
6451
6519
6587
6655
6723
6790
68
641
80 6858
6926
6994
7061
7129
7197
7264
7332
7400
7467
68
642
80 7535
7603
7670
7738
7806
7873
7941
8008
8076
8143
68
643
80 821 1
8279
8346
8414
8481
8549
8616
8684
8751
8818
67
644
80 8886
8953
9021
9088
9156
9223
9290
9358
9425
9492
67
645
80 9560
9627
9694
9762
9829
9896
9964
• • • • •
67
^l
81
0031
0098
0165
67
646
81 0233
0300
0367
0434
0501
0569
0636
0703
0770
0837
67
^7
81 0904
0971
1039
1 106
"73
1240
1307
1374
144 1
1508
67
648
81 1575
1642
1709
1776
1843
1910
1977
2044
2111
2178
67
649 1 8x 2245
2312
2379
2445
2512
2579
2646
2713
2780
2847
67
650
81 2913
2980
3047
3'i4
3181 i 3247
3314
3381
3448
3514
67
651
8x 3581
3648
3714
3781
3848 ! 3914
3981
4048
4114
4181
67
652
8x 4248
43 >4
4381
4447
45H 4581
4647
4714
4780
4847
67
J53
81^4913
4980
5046
5"3
5179 5246
5312
5378
5445
55"
66
654
655
81 5578
5644
57"
5777
5843 , 5910
5976
6042
6109
6175
66
81 6241
6308
6374
6440
6506
6573
6639
6705
6771
6838
66
656
81 6904
6970
7036
7102
7169
7235
7301
7367
7433
7499
66
^l
81 7565
7631
7698
7764
7S30
7896
7962
8028
8094
8160
66
658
81 8226
8292
8358
8424
8490
8556
8622
8688
8754
8820
66
659 81 8885
8951
9017
9083
9149
9215
9281
9346
9412
9478
66
660
81 9544
9610
9676
9741
9807
9873
9939
66
660
82
0004
0070
0136
66
661
82 0201
0267
0333
09S9
0399
0464
0530
0595
0661
0727
0792
66
662
82 0S58
0924
1055
1 120
1 186
1251
1317
1382
1448
66
5^3
8a 15 14
1579
1645
1710
1775
184E
1906
1972
2037
2103
65
664
82 2168
2233
2299
2364
2430
2495
2560
2626
2691
2756
65
6^5
82 2822
2887
2952
3018
3083
3148
3213
3279
3344
3409
65
666
82 3474
3539
3605
3670
3735
3800
3865
3930
3996
4061
55
/ 667 ' 83 4126
4191
4256
4321
4386
4451
45 '6
4581
4646
47"
^J
668
82 4776
4841
4906
4971
5036
5101
5166
5231
•
5296
5361
65
D
N :
I
2
3
4
5
6
7
8
9
52
MATHEMATICAL TABLES.
N
669
I
2
3
4
S
6
7
8
9
D
65
82 5426
5491
5556
5621
5686
5751
5815
5880
5945
6010
670
82 6075
6140
6204
6269
6334
6399
6464
6528
6593
6658
65
671
82 6723
6787
6852
6917
6981
7046
7111
717s
7240
7305
65
672
82 7369
7434
7499
7563
7628
7692
7757
7821
7886
8531
7951
65
673
82 8015
8080
8144
8209
8273
8918
8338
8402
8467
8595
64
674
82 8660
8724
8789
8853
8982
9046
9111
9175
9239
64
676
82 9304
9368
9432
9497
9561
9625
9690
9754
9818
9882
64
676
82 9947
64
676
83
001 1
0075
0139
0204
0268
0332
0396
0460
0525
64
677
83 0589
obSS
0717
0781
084s
0909
0973
1037
X102
1166
64
678
83 1230
1294
1358
1422
X486
15 JO
1614
1678
1742
1806
64
679
83 1870
1934
1998
2062
2126
2189
2253
2317
2381
2445
64
680
83 2509
2573
2637
2700
2764
2828
2892
2956
3020
308^
64
681
83 3147
3211
3275
3338
3402
3466
3530 
3593
3657
3721
64
682
83 3784
3848
3912
3975
4039
4103
4166
4230
4294
4357
64
683
83 4421
4484
4548
461 1
4675
4739
4802
4866
4929
4993
64
684
83 5056
5120
5183
5247
5310
5373
5437
5500
5564
5627
63
685
83 5691
5754
5817
5881
5944
6007
6071
6134
6197
6261
63
686
83 6324
6387
6451
7083
6514
6577
6641
6704
6767
6830
6894
63
687
83 6957
7020
7146
7210
7273
7336
7399
7462
7525
63
688
83 75»8
7652
7715
7778
7841
7904
7967
8030
8093
8156
63
689
83 8219
8282
8345
8408
8471
8534
8597
8660
8723
8786
63
690
83 8849
8912
8975
9038
9101
9164
9227
9289
9352
9415
63
691
83 9478
9541
9604
9667
9729
9792
9855
9918
9981
63
691
84
• • • •
0043
63
692
84 0106
0169
0232
0294
0357
0420
0482
0545
0608
0671
63
693
84 0733
0796
0859
0921
0984
1046
1 109
1172
1234
1297
63
694
84 1359
1422
1485
1547
1610
1672
1735
1797
i860
1922
63
695
84 1985
2047
21 10
2172
2235
2297
2360
2422
2484
2547
62
696
84 2609
2672
2734
2796
28^9
2921
2983
3046
3108
3170
62
697
84 3233
3295
3357
3p8o
3420
3482
3544
3606
3669
3731
3793
62
698
84 3855
3918
4042
4104
4166
4229
4291
4353
4415
62
699
84 4477
4539
4601
4664
4726
4788
4850
4912
5532
4974
5594
5036
62
700
84 5098
5160
5222
5284
5346
5408
5470
5656
62
701
84 5718
S780
5842
5904
5966
6028
6090
6151
6213
6275
62
702
84 6337
6399
6461
6523
6585
6646
6708
6770
6832
6894
62
703
84 6955
7017
7079
7141
7202
7264
7326
7388
7449
7511
62
704
84 7573
7634
7696
7758
7819
7881
7943
8004
8066
8128
62
705
84 8i<89
8251
8312
8374
8435
8497
8559
8620
8682
8743
9358
62
706
84 8805
8866
8928
8989
9051
9112
9174
9235
9297
61
707
84 9419
9481
9542
9604
9665
9726
9788
9849
9911
9972
61
708
85 0033
0095
0156
0217
0279
0340
0401
0462
0524
0585
61
709
85 0646
0707
0769
0830
0891
0952
1564
1014
1075
1 136
1 197
61
710
85 1258
1320
1381
1442
1503
1625
1686
1747
1809
61
711
85 1870
1931
1992
2053
21 14
2175
2236
2297
2358
2419
61
712
85 2480
2541
2602
2663
2724
2785
2846
2907
2968
3029
61
713
85 3090
3150
3211
^iP
3333
3394
3455
3516
3577
3637
61
714
N
85 3698
3759
3820
3881
3941
4002
4063
4124
4185
•
4245
61
D
I
2
3
' 4
s
6
7
8
9
I'
/
I.OGARITHMS OF NUMBERS.
8
43^7 44^S 4488 4549
4974 5054 509s 5156
5580 5640 5701 5761
6185 624s ^306 6366
678g 68so 6910 6970
4610 4670 4731
5216 5277 5337
5822 5882 594J
6427 6487 6548
7031 7091 7152
72' siS^^ 7J93 7453
^^3 It !p7 8597 86S7
7M I d. ^^^^ 9799 9S59
75^3
Si 16
871S
93^1
99^^
7574
8176
8778
9379
9978
86^
86^
7634
8236
8838
9439
7694 7755
8297 8357
8898 8958
9499 9559
7815 7875
8417 8477
9018 9078
9619 9679
0038 0098 0158 0218 0278
86. i?^ 0996 lOSS
^^^^ '594 les^
EE mO
X7'4.
^3x0
0578
1 176
1773
2370
2966
0637 0697 0757
1236 1295 1355
1833 1893 1952
2430 2489 2549
3025 3085 3144
0817 0877
H15 1475
2012 2072
2608 2668
3204 3263
7J4/S^'^^
sS^c^ A^3<^
9525 9584 964^ 9701 9760
01 I I 0170 0228 0287 0345
0696 0755 0813 0872 0930
1281 1339 1398 1456 I5I5
1865 1923 I98I 2040 2098
S3
D
4792 4852 61
5398 5459 61
6003 6064 61
6608 6668 60
7212 7272 60
60
60
60
60
60
60
60
60
60
60
60
3620 3680 3739 3799 3858 59
4214 4274 4333 4392 4452 59
4808 4867 4926 4985 5045 59
5400 5459 5519 5578 5637 59
5992 6051 61x0 6169 6228 59
6583 6642 6701 6760 6819 59
7173 7232 7291 7350 7409 59
7762 7821 7880 7939 7998 59
8350 8409 8468 8527 8586 59
8938 8997 9056 9114 9173 59
59
59
59
58
58
2448 2506 2564 2622 2681 58
3030 3088 3146 3204 3262 58
361 1 3669 3727 3785 3844 58
4192 4250 4308 4366 4424 58
4772 4830 4888 4945 5003 58
5351 5409 5466 5524 5582 58
5929 5987 6045 6102 6160 58
6507 6564 6622 6680 6737 58
7083 7141 7199 7256 7314 58
7659 7717 7774 7832 7889 58
8234 8292 8349 8407 8464 57
8809 8866 8924 8981 9039 57
9383 9440 9497 9555 9612 57
9956 ••• 57
0013 0070 0127 0185 57
0528 0585 0642 0699 0756 57
«; 6 7 8 Q D
54
MATHEMATICAL TABLES.
N
8
D
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
88 0S14
88 1385
88 1955
88 2525
88 3093
0871
1442
2012
2581
3150
0928
1499
2069
2638
3207
0985
1556
2126
2695
3264
1042
1613
2183
2752
3321
1099
1670
2240
2809
3377
1156
1727
2297
2866
3434
1213
1784
2354
2923
3491
1271
1 841
241 1
2980
3548
1328
1898
2468
3037
3605
88 3661
88 4229
88 4795
88 5361
88 5926
3718
4285
4852
5418
5983
3775
4342
4909
5474
6039
3832
4399
4965
5531
6096
3888
4455
5022
5587
6152
3945
4512
5078
5644
6209
4002
4569
5135
5700
6265
4059
4625
5192
5757
6321
4115
4682
5248
581
637
4172
4739
5305
5870
6434
88 6491
88 7054
88 7617
88 8179
88 8741
6547
7111
7674
8236
8797
6604
7167
7730
8292
8853
6660
7223
7786
8348
8909
6716
7280
7842
8404
8965
6773
7336
7898
8460
9021
6829
7392
8516
9077
6885
7449
801 1
8573
9134
6942
7505
8067
8629
9190
6998
7561
8123
'8685
9246
57
57
57
57
57
57
57
57
57
56
56
56
56
56
56
775
776
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
794
805
N
88 9302
88 9862
89
89 0421
89 0980
89 1537
9358
9918
0477
1035
1593
9414
9974
0533
1091
1649
9470 9526
OOJO
0589
1 147
1705
0086
0645
1203
1760
9582 9638 9694 9750 9806
0I4I
0700
1259
I8I6
0197
0756
1314
1872
0253
0812
1370
1928
0309
0868
1426
1983
0365
0924
1482
2039
89 209s
89 2651
89 3207
89 3762
89 4316
2150
2707
3262
3817
4371
2206
2762
3318
3873
4427
2262
2818
3373
3928
4482
2317
2873
3429
3984
4538
2373
2929
3484
4039
4593
2429
2985
3540
4094
4648
2484
3040
3595
4150
4704
2540
3096
3651
4205
4759
2595
3>5i
3706
4261
4814
89 4870
89 5423
89 5975
89 6526
89 7077
4925
6030
6581
7132
4980
5533
6085
6636
7187
5036
5588
6140
6692
7242
5091
5644
6195
6747
7297
5146
5699
6251
6802
7352
5201
5754
6306
6857
7407
5257
5809
6361
6912
7462
5312
5864
6416
6967
7517
5367
5920
6471
7022
7572
8g 7627
89 8176
89 8725
89 9273
89 9821
90
7682
8231
8780
9328
9875
7737
82S6
8835
9383
9930
7792
8341
8890
9437
9985
7847
8396
8944
9492
0039
795 90 0367 0422 0476 0531 0586
796 90 0913 0968 1022 1077 1 131
797 90 1458 1513 1567 1622 1676
798 90 2003 2057 21 12 2166 2221
799 90 2547 2601 2655 2710 2764
7902
8451
8999
9547
7957
8506
9054
9602
8012
8561
9109
9656
8067
8615
9164
9711
8122
8670
9218
9766
0094 0149 0203 0258 0312
0640
1186
1731
2275
2818
0695
1240
1785
2329
2873
0749
1295
1840
2384
2927
0804
1349
1894
2438
2981
0859
1404
1948
2492
3036
800 90 3090 3144 3199 3253 3307
801 90 3633 3687 3741 3795 3849
802 90 4174 4229 4283 4337 4391
803 90 4716 4770 4824 4878 4932
804 90 5256 5310 5364 5418 5472
90 5796 5850 5904 5958 6012
3361
39<H
4445
4986
5526
3416
3958
4499
5040
5580
3470
4012
4553
5094
5634
3524
4066
4607
5148
5688
3578
4120
4661
5202
5742
56
56
56
56
56
56
56
55
55
55
55
55
55
ii
55
55
55
55
55
55
55
55
54
54
54
54
54
54
54
54
6066 6119 6173 6227 6281 54
^ 6 7 S I)
LOGARITHMS OF NUMBERS.
55
N
8
D
'So6
■807
808
;809
90 6335 6389
90 6874 6927
90 7411 7465
90 7949 8002
6443 6497 6551
6981 7035 7089
75*9 7573 7626
S056 81 10 8163
6604
7143
7680
8217
6658
7196
7734
8270
6712
7250
7787
8324
6766
7304
7841
8378
6820
7358
7«95
8431
810
811
812
8f2
814
90 8485 8539
9& 9021 9074.
90 9556 9610
y*^ «>•••• m m 9 m m m
91 0091 0144.
91 0624 0678
8592 8646
9128 9181
9663 9716
0197 0251
0731 0784
8699
9235
9770
0304
0838
8753
9289
9823
0358
0891
8807
9342
9877
041 1
0944
8860
9396
9930
0464
0998
8914
9449
9984
0518
105 1
8967
9503
0037
0571
1104
54
54
54
54
54
54
54
53
53
53
815
816
817
:8ig
819
91 1158 121 1 1264.
911690 1743 '797
91 2222 2275 ^3^^
91 2753 2806 2»59
9» 3284 3337 3390
1317
1850
2381
29<3
3443
1371
1903
2435
2966
3496
1424
1956
2488
3019
3549
1477
2009
2541
3072
3602
1530
2063
2594
3125
3655
1584
2116
2647
3178
3708
1637
2169
2700
3231
3761
820
821
822
823
824
91
91
91
91
91
3814
4343
4872
S40O
5927
3867
439^
4925
5453
5980
3920
4449
4977
6033
3973
4502
5030
5558
6085
4026
4555
5083
561 1
6138
4079
4608
5136
5664
6191
4132
4660
5189
5716
6243
4184
4713
5241
5769
6296
4237
4766
5294
5822
6349
4290
4819
5347
5875
6401
825
826
827 i
828
829
91
91
9X
91
91
6454
6980
7506
8030
8555
6507
7033
7558
8083
8607
6559
7085
761 1
8x35
8659
6612
7138
7663
8188
8712
6664
7190
7716
8240
8764
6717
7243
7768
8293
8816
6770
7295
7820
8345
8869
6822
7348
7873
8397
8921
6S75
7400
7925
8450
8973
6927
7453
7978
8502
9026
830 ! 9X9078 9130
831 91 9601 9653
831 92
832 92 0123 0176
833 92 0645 0697
834 9a 1166 1218
9183
9706
0228
0749
1270
9235
9758
0280
0801
1322
9287
9810
0332
0853
1374
9340
9862
0384
0906
1426
9392
9914
0436
0958
1478
9444
9967
0489
lOIO
1530
9496 9549
0019
0541
1062
1582
0071
0593
1 1 14
1634
835
836
838
839
92 1686
92 2206
9a 2725
92 3244.
92 3762
1738
2258
^777
3296
3814
1790
2^10
2829
m
1842
2362
2881
3399
3917
1894
2414
2933
3451
3969
1946
2466
2985
3503
4021
1998
2518
3037
3555
4072
2050
2570
3089
3607
4124
2102
2622
3140
3658
4176
2154
2674
3192
3710
4228
840
841
842
,843
' 844
9» 4279
92 4796
92 5312
92 5828
92 6342
4848
53^4
5879
6394
4383
4899
5415
593 «
6445
4434
4951
5467
5982
6497
4486
5003
5518
6034
6548
4538
5054
5570
6p85
6600
4589
5106
5621
6137
6651
4641
5157
5673
6188
6702
4693
5209
5725
6240
6754
4744
5261
5776
6291
6805
845
1846
I 847
'848
I 849
92 6857
92 7370
92 78S3
92 8396
92 8908
6908
7422
7935
8447
8959
6959
7473
7986
8498
9010
701 1
7524
8037
8549
9061
7062
7576
8088
8601
9112
7114
7627
8140
8652
9163
7165
7678
8191
8703
9215
7216
7730
8242
8754
9266
7268
7781
8293
8805
9317
73^9
7832
8345
8857
9368
93 9419
92 9930
9470
9981
9521 9572 9623
9674 9725 9776 9827 9879
7
8
53
53
53
53
53
53
53
53
53
53
53
53
52
52
52
52
52
52
52
52
52
52
52
52
52
52
52
52
52
51
51
51
51
51
51
51
51
51
D
S6
MATHEMATICAL TABLES.
N
I
2
3
4
5
6
7
8 9
D
851
93
0032
0083
0134
0185
0236
0287
0338 0389
0S47 0898
51
852
93 0440
0491
0542
0592
0643
0694
0745
0796
51
853
93 0949
IOCX>
105 1
1 102
1153
1203
1254
1305
1356 1407
1865 1915
SI
854
93 1458
1509
1560
1610
1661
1712
1763
1814
51
51
856
93 1966
2017
2068
2118
2169
2220
2271
2322
2372 2423
856
93 2474
2524
2575
2626
2677
2727
2778
2829
2879 2930
SI
857
93 2981
3031
3082
3133
3183
3234
3285
3335
3386 3437
3892 3943
4397 4448
SI
858
93 3487
3538
3589
3639
3690
3740
3791
3841
51
859
93 3993
4044
4094
4145
4195
4246
4296
4347
SI
860
93 4498
4549
4599
4650
4700
4751
4801
4852
4902 4953
50
861
93 5003
5054
5104
5154
5205
5255
5306
5809
5356
5406 5457
50
862
93 5507
5558
5608
5658
5709
5759
5860
5910 5960
50
863
93 601 I
6061
6111
6162
6212
6262
6313
6815
6363
6865
6413 6463
50
864
93 6514
6564
6614
6665
6715
6765
6916 6966
50
865
93 7016
7066
7117
7618
7167
7217
7267
7317
7367
7418 7468
50
866
93 75>8
7568
7668
7718
7769
8269
7819
7869
7919 7969
50
867
93 8019
8069
8119
8169
8219
8319
8370
8420 8470
50
868
93 8520
8570
8620
8670
8720
8770
8820
8870
8920 8970
so
869
93 9020
9070
9120
9170
9220
9270
9320
9369
9419 9469
50
870
93 9519
9569
9619
9669
9719
9769
9819
9869
9918 9968
50
871
94 0018
0068
0118
0168
0218
0267
0317
0815
0367
0865
0417 0467
50
872
94 0516
0566
0616
0666
0716
0765
0915 0964
50
873
94 1014
1064
1 1 14
1 163
1213
1263
1809
1362
1412 1462
50
874
94 1511
1561
1611
1660
1710
1760
1859
1909 1958
50
875
94 2008
2058
2107
2157
2207
2256
2306
2801
2355
2405 2455
50
876
94 2504
2554
2603
2653
2702
2752
2851
2901 2950
50
^77
878
94 3000
3049
3099
3148
3198
3247
3297
3346
3396 3445
49
94 3495
3544
408!;
3^43
3692
3742
3791
3841
3890 3939
49
879
94 3989
4038
4137
4186
4236
4285
4335
4384 4433
49
880
94 4483
4532
4581
4631
4680
4729
4779
4828
4877 4927
49
881
94 4976
5025
5074
5124
5>73
5222
5272
5321
5370 5419
5862 59 r2
49
882
94 5469
5518
5567
5616
5665
5715
5764
5813
49
883
94 5961
6010
6059
6108
6157
6207
6256
6305
6354 6403
6845 6894
49
884
94 6452
6501
6551
6600
6649
6698
6747
6796
49
49
885
94 6943
6992
7041
7090
7140
7189
7238
7287
7336 7385
886
94 7434
7483
7532
7581
7630
7679
7728
7777
7826 7875
49
887
94 7924
7973
8022
8070
8119
8168
8217
8266
8315 8364
8804 8853
49
888
94 8413
8462
8511
8560
8609
8657
8706
8755
49
889
94 8902
8951
8999
9048
9097
9146
9195
9244
9292 9341
49
890
94 9390
9439
9488
9536
9585
9634
9683
9731
9780 9829
49
891
94 9878
9926
9975
49
891
95
0024
0073
OI2X
0170
0219
0267 0316
0754 0803
49
892
95 0365
95 0851
0414
0462
0511
0560
0608
0657
0706
49
893
0900
0949
0997
1046
1095
"43
1 192
1240 1289
49
894
95 1338
1386
1435
1483
1532
1580
1629
1677
1726 1775
49
896
95 182J
^ 95 2308
1872
1920
1969
2017
2066
21 14
2163
22 I I 2260
48
896
2356
^?5
2453
2502
2550
2599
2647
2696 2744
48
897
N
95 2792
2841
2889
2938
2986
3034
3083
3131
3180 3228
48
I
2
3
4
5
6
7
8 9
D
LOGARITHMS OF NUMBERS.
57
i N
8
D
48
48
48
48
48
48
48
898 95 3276 3325 3373 3421 3470
899 ! 95 3760 3808 3856 3905 3953
3518
4001
3566
4049
361.
409^
3663
4146
37"
4194
900 ' 95 4243 4291 4339
901 I 95 4725 4773 4821
95 5207 5255 5303
95 5688 5736 5784
95 6168 6216 6265
902
903
904
905
906
,907
I 908
909
4387
4869
5351
32
313
4435
4918
5399
5880
6361
95 6649
95 7128
95 7607
95 8086
95 8564
6697
7176
7655
8134
8612
6745
7224
7703
8181
8659
6793
7272
7751
8229
8707
6840
7320
7799
8277
8755
4484
4966
5447
5928
6409
4532
5014
5495
5976
6457
4580
5062
5543
6024
6505
4628
5110
5592
6072
6553
4677
5158
5640
6120
6601
6888
7368
7847
8325
8803
6936
7416
7894
8373
8850
6984
7464
7942
8421
88q8
7032
7512
7990
8468
8946
7080
7559
8038
8516
8994
48
48
48
48
48
48
48
48
48
48
47
910
911
! 912
912
913
914
95 9041
95 9518
959995
96
96 0471
96 0946
9089
9566
0042
0518
0994
9137
9614
0090
0566
1041
9185
9661
0138
0613
1089
9232
9709
0185
066 X
1136
9280
9757
0233
0709
1184
9328
9804
• • • «
0280
0756
1231
9375
9852
0328
0804
1279
9423
9900
0376
0851
1326
9471
9947
Q423
0899
1374
916 I 96 1421
916
917
918
919
96 1895
96 2369
96 2843
9& 33^6
1469
1943
2417
2890
3363
1516
1990
2464
2937
3410
1563
2038
251 1
2985
3457
1611
2085
2559
3032
3504
1658
2132
2606
3079
3552
1706
2180
2653
3126
3599
1753
2227
2701
3174
3646
1801
2275
2748
3221
3693
1848
2322
2795
3268
3741
47
47
47
47
47
920
921
922
923
924
963788
96 4260
964731
96 5202
96 5672
3835
4307
4778
5249
5719
3882
4354
4825
5296
5766
3929
4401
4872
5343
5813
3977
4448
4919
5390
5^
4024
4495
4966
5437
5907
4071
4542
5013
5484
5954
4118
4590
5061
5531
6001
4165
4637
5108
6048
4212 i 47
4684
5155
5625
6095
47
47
47
47
925
926
927
928
929
930
1931
' 932
933
933
.934
i"
985
936
937
938
939
940
941
942
943
IT
96 6142
96 661 I
96 7080
96 754S
96 8016
6189
6658
7127
7595
8062
6236
6705
7173
7642
8109
6283
6752
7220
7688
8156
6329
6799
7267
7735
8203
6376
6845
7314
7782
8249
6423
6892
7361
7829
8296
6470
6939
7408
7875
8343
6517
6986
7454
7922
8390
6564
7033
7501
7969
8436
968483
96 8950
96 9416
96 9882
97
97 0347
8530
8996
9463
9928
8576
9043
9509
9975
0393 0440
8623
9090
9556
0021
0486
8670
9136
9602
0068
0533
8716
9183
9649
01 14
0579
8763
9229
9695
0161
0626
8810
9276
9742
0207
0672
8856
9323
9789
0254
0719
8903
9369
9835
0300
0765
97 0812
97 1276
97 1740
97 2203
97 2666
0858
1322
1786
2249
2712
0904
1369
1832
2295
2758
0951
1415
1879
2342
2804
0997
1461
1925
2388
2851
1044
1508
1971
2434
2897
1090
>554
2018
2481
2943
"37
1601
2064
2527
2989
1x83
1647
21 10
2573
3035
97 3128
97 3590
97 4051
97 4512
3174
3636
4097
4558
3220
3682
4143
4604
3266
3728
4189
4650
3313
3774
4235
4696
3359
3820
4281
4742
3405
3866
4327
4788
3451
3913
4374
4834
3497
3959
4420
4880
8
1229
1693
2157
2619
3082
3543
4005
4466
4926
47
47
47
47
47
47
47
47
47
47
46
46
46
46
46
46
46
46
46
ii
D
 "It. '/*f /■
• 58
MATHEMATICAL TABLES.
N
944
I
2
3
4
5
6
7
8
9
D
46
97 4972
5018
5064
5110
5156
5202
5248
5294
5340
5386
945
97 5432
5478
5524
5570
5616
5662
5707
5753
•
5799
5845
46
946
97 5891
5937
5983
6029
6075
6121
6167
6212
6258
6304
46
947
97 6350
97 6808
6396
6442
64S8
6533
6579
6625
6671
6717
6763
46
948
6854
6900
6946
6992
7037
7083
7129
7175
7220
46
949
97 7266
7312
7358
7403
7449
7495
7541
7586
7632
7678
46
960
97 7724
7769
7815
7861
7906
7952
8409
7998
8454
8043
8089
8135
46
951
97 8i8i
8226
8272
8317
8363
8500
8546
8591
46
952
97 8637
8683
8728
8774
8819
8865
891 1
8956
9002
9047
46
953
979093
9138
9184
9230
9275
9321
9366
9412
9457
9503
46
954
97 9548
9594
9639
9685
9730
9776
982 X
9867
9912
9958
46
966
98 0003
0049
0094
0140
D185
0231
0276
0322
0367
082 X
^l^
45
956
98 0458
0503
0549
0594
0640
0685
0730
0776
0867
45
957
98 0912
0957
1003
1048
1093
"39
1x84
1229
1275
1320
45
958
98 1366
X411
1456
1 501
1547
1592
1637
1683
1728
1773
45
959
98 1819
1864
1909
1954
2000
2Q45
2090
2135
2181
2226
45
960
98 2271
2316
2362
2407
2452
2497
2543
2588
2653
3085
2678
45
961
98 2723
2769
2814
2859
2904
2949
2994
3040
3130
45
962
98 3175
3220
3265
33 JO
3356
3807
3401
3446
3491
3536
3581
45
963
98 3626
3671
3716
3762
3852
3897
3942
3987
40J2
45
964
98 4077
4122
4167
4212
4257
4302
4347
4392
4437
4482
45
965
98 4527
4572
4617
4662
4707
4752
4797
4842
4887
4932
45
966
98 4977
5022
5067
5112
5157
5202
5247
5292
5337
5382
45
967
98 5426
5471
5516
5561
5606
5651
5696
5741
5786
5830
45
968
98 5875
5920
5965
6010
6055
6100
6144
6189
6234
6279
45
969
98 6324
6369
6413
6458
6503
6548
6593
6637
6682
6727
45
970
98 6772
6817
6S61
6906
6951
6996
7040
7085
7130
7175
45
971
98 7219
7264
7309
7353
7800
7398
7443
7488
7532
7577
7622
45
972
98 7666
77"
7756
7845
7890
7934
7979
8024
8068
45
973
98 81 13
8157
8202
8247
8291
8336
838X
8425
8470
8514
45
974
98 8559
8604
8648
8693
8737
8782
8826
8871
8916
8960
45
975
98 9005
9049
9094
9138
9185
9227
9272
9316
9361
9405
45
976
98 9450
9494
9539
9583
9628
9672
9717
9761
9806
9850
44
977
98 9895
9939
9983
• • •
44
977
99
0028
0072
0117
0161
0206
0250
0294
44
978
99 0339
0383
0428
0472
0516
0561
0605
0650
0694
0738
44
979
99 0783
0827
0871
0916
0960
1004
1049
1093
1137
1 182
44
980
99 1226
1270
I3J5
1359
1403
1448
1492
1536
1580
1625
44
981
99 1669
1713
1758
1802
1846 ! 1890
1935
1979
2023
2067
44
982
99 21 I I
2156
2200
2244
2288
2333
2377
2421
2465
2509
44
983
99 2554
99 2^95
2598
2642
2686
2730
2774
2819
2863
2907
2951
44
984
3039
3083
3127
3172
3216
3260
3304
3348
3392
44
985
99 3436
3480
3524
3568
3613
3657
3701
3745
3789
3833
1
44
986
99 3877
3921
3965
4009
4053
4097
4141
4185
4229
4273
44
987
99 4317
4361
4405
4449
4493
4537
4581
4625
4669
4713
44
988
99 4757
4801
4845
4889
4933
4977
5021
5065
5108
5152
44
989
N
99 5196
5240
5284
5328
5372
5416
5460
5504
5547
5591
44
D
I
2
3
4
5
6
7
8
9
LOGARITHMS OF NUMBERS.
59
N
I
2
3
4
5
6
7
8
9
D
990
9^ 5635
5679
5723
5767
5811
5854
5898
5942
5986
6030
44
991
9^ 6074.
6117
6161
6205
6249
6293
6337
6380
6424
6468
44
992
99 6512
6555
6599
6643
6687
6731
6774
6818
6862
6906
44
993
9Q 694.9
6993
7037
7080
7124
7168
7212
7255
7299
7343
44
994
99 73S^
7430
7867
7474
7910
7517
7954
7561
7998
7605
8041
7648
7692
7736
7779
44
44
995
99 7823
8085
8129
8172
8216
996
99 8259
8303
8347
83QO
8434
8477
8521
8564
8608
8652
9087
44
997
99 8695
8739
8782
8826
8869
8913
8956
9000
9043
44
99»
99 9131
9174
9218
9261
9305
93*8
9392
9435
9479
9522
44
999
99 9565
9609
9652
9696
9739
9783
9826
9870
9913
9957
43
D
N
I
2
3
4
5
6
7
8
9
/
6o
MATHEMATICAL TABLES.
TABLE No. IL— HYPERBOLIC LOGARITHMS OF NUMBERS
FROM 1. 01 TO 30.
Number.
Logarithm.
Number.
Logarithm.
Number.
Logarithm.
Number.
Logarithm.
I.OI
1.02
1.03
1.04
1.05
.0099
.0198
.0296
.0392
.0488
1.36
137
1.38
139
1.40
•3075
.3148
.3221
.3293
•3365
I.7I
1.72
173
1.74
1 1.75
•5365
.5423
.5481
•5539
•5596
2.06
2.07
2.08
2.09
2.10
.7227
•7275
•7324
•7372
.7419
1.06
1.07
1.08
1.09
1. 10
.0583
.0677
.0770
.0862
■0953
1.41*
1.42
143
1.44
1.45
.3436
.3507
.3577
.3646
•3716
1.76
1.77
1.78
1.79
1.80
.5653
.5710
.5766
.5822
.5878
2. II
2.12
2.13
2.14
2^15
.7467
•7514
.7561
.7608
.7655
1. 11
1. 12
^•13
1. 14
^•15
.1044
.1133
.1222
.1310
.1398
1.46
1.47
1.48
1.49
1.50
.3784
.3853
.3920
.3988
•4055
1.81
1.82
1.83
1.84
1.85
.5933
.5988
.6043
.6098
.6152
2.16
2.17
2.18
2.19
2.20
.7701
.7747
•7793
.7839
.7885
1. 16
1.17
1. 18
1. 19
1.20
.1484
.1570
.1655
.1740
.1823
I5I
152
153
1^54
1.55
.4121
.4187
•4253
.4318
■4383
1.86
1.87
1.88
1.89
1.90
.6206
.6259
.6313
.6366
.6419
2.21
2.22
2.23
2.24
2.25
•7930
.7975
.8020
.8065
.8109
1. 21
1.22
1.23
1.24
1.25
.1906
.1988
.2070
.2151
.2231
1.56
157
1.58
1.59
1.60
.4447
.4511
•4574
•4637
.4700
1.91
1.92
1.93
1.94
1.95
.6471
•6523
•6575
.6627
.6678
2.26
2.27
2.28
2.29
2.30
.8154
.8198
.8242
.8286
•8329
1.26
1.27
1.28
1.29
1.30
.2311
.2390
.2469
.2546
.2624
1.61
1.62
1.63
1.64
1.65
.4762
.4824
.4886
.4947
.5008
1.96
1.97
1.98
1.99
2.00
.6729
.6780
.6831
.6881
.6931
2.31
2.32
2.33
2.34
2.35
.8372
.8416
.8458
.8502
.8544
131
1.32
1.33
1.34
I.3S
.2700
.2776
.2852
.2927
.3001
1.66
1.67
1.68
1.69
1.70
.5068
.5128
.5188
.5247
.5306
2.01
2.02
2.03
2.04
2.05
.6981
•7031
.7080
.7129
.7178
2.36
2.37
2.38
2.39
2.40
.8587
.8629
.8671
•8713
.8755
j
HYPERBOLIC LOGARITHMS OF NUMBERS
6l
1
4
!f umber.
Logarithm.
Number.
Logarithm.
Number.
Logarithm.
1
Number.
Logarithm.
2.41
.8796
2.8l
1.0332
3.21
I.1663
3.61
1.2837
2.42
.8838
2.82
1.0367
3.22
1. 1694
3.62
1.2865
2.43
.8879
2.83
1.0403
3.23
I.I725
3.63
1.2892
2.44 .8920
2.84
1.0438
3.24
I.1756
364
1.2920
2.45 .8961
2.85
1.0473
3.25
I.I787
3.65
1.2947
2.46 .9002
2.86
1.0508
3.26
1.1817
3.66
1.2975
2.47 .9042
2.87
10543
327
1. 1848
3.67
1.3002
2.48 1 9083
2.88
1.0573
328
1. 1878
3.68
1.3029
2.49 1 .9123
2.89
I.0613
3.29
1. 1 909
3.69
1.3056
2.50 9163
2.90
1.0647
3.30
11939
370
1.3083
2.51
.9203
2.91
1.0682
3.31
1. 1969
3.71
1.3110
25*
9243
2.92
1.0716
3.32
1. 1999
372
I.3137
*53
.92^2
2.93
1.0750
333
1.2030
3.73
I.3164
2.54
.9322
2.94
1.0784
334
1.2060
3.74
I.319I
255
.9361
2.95
i.o8i8
3.35
1.2090
3.75
1.3218
2.56
.9400
2.96
1.0852
3.36
1.2119
3.76
1.3244
2.57
•9439
2.97
1.0886
337
1. 2149
3.77
I.3271
2.58
.9478
2.98
1.0919
3.38
I.2179
3.78
1.3297
1 2.59
.9517
2.99
1.0953
339
1.2208
3.79
1.3324
2.60
9555
3.00
1.0986
3.40
1.2238
3.80
13350
2.61
9594
3.01
1.1019
3.41
1.2267
3.81
1.3376
2.62
.9632
3.02
1. 1053
3.42
1.2296
3.82
1.3403
2.63
.9670
303
1. 1 086
343
1.2326
383
1.3429
2.64
.9708
304
1.1119
344
1.2355
3.84
13455
2.65
.9746
305
1.1151
3.45
1.2384
3.85
1.3481
2.66
.9783
3.06
1.1184
3.46
1.2413
3.86
13507
2.67
.9821
307
1.1217
3.47
1.2442
3.87
13533
2.68
.9858
3.08
1. 1249
3.48
1.2470
3.88
1.3558
1 269
.9895
.309
1.1282
349
1.2499
3.89
13584
2.70
9933
3.10
1.1314
350
1.2528
3.90
1.3610
2.71
.9969
3"
1. 1346
3.51
1.2556
391
1.3635
2.72 1.0006
3.12
1.1378
352
1.2585
3.92
1.3661
2.73 1.0043
313
1.1410
3.53
1. 2613
3.93
1.3686
2.74
1.0080
3.14
1.1442
3.54
1. 2641
3.94
1.3712
2.75
1.0116
315
1.1474
3.55
1.2669
3.95 .
1.3737
^ 2.76
10152
3.16
1. 1506
3.56
1.2698
3.96
1.3762
2.77
1.0188
317
1.1537
3.57
1.2726
397
1.3788
2.78
1.0225
3.18
1.1569
3.58
1.2754
3.98
1.3813
2.79 1.0260
319
1. 1600
359
1.2782
399
1.3838
2.80
1.0296
3.20
1.1632
3.60
1.2809
4.00
1.3863
!
MATHEMATICAL TABLES.
Logiiithm.
I Numbe.
Logarithm.
Numbn.
L.«lfithlB.
Numbn.
l.:g..ilhn.
1.3888
4.41
1.4839
4.8.
15707
S2I
.6506
'■39'3
1 442
..4861
4.82
15728
S22
16525
'■3938
i 443
r.4884
4.83
15748
523
.65.4
1.3962
; 444
1.4907
4.84
1.5769
S24
.6563
13987
,4.45
1.4929
4.85
1.5790
525
.6582
1. 401 2
, 4.46
1495'
4.86
1.5S.0
526
.6601
1.4036
! 447
1.4974
4.87
..583.
527
16620
..4o6r
4.48
1.4996
4.88
15851
5.28
16639
1.408s
4.49
I.50I9
4.89
..5872
5.29
16658
1.4110
1 4.50
1.5041
4.90
1.5892
5.30
16677
1.4134
! 451
1.5063
4.91
15913
531
16696
i4'S9
; 4.52
1.5085
4.92
15933
S32
I67I5
1.4183
i 453
1.5107
4.93
15953
533
16734
1.4207
454
1.5129
494
15974
S34
16752
1.4231
4.55
1.5151
4.95
15994
535
16771
I42SS
i 4.56
i.S"73
4.96
16014
536
16790
1.4279
, 4.57
'■S'95
4.97
16034
S37
16808
1.4303
,4.58
15217
4.98
16054
538
16827
14327
1 459
1.5239
499
16074
539
16845
I43S1
I4.60
1526.
5.00
16094
540
I6S64
•■4375
, 4.61
1.5282
! 501
16.14
S41
16882
1.4398
4.62
15304
5.02
16134
542
1690,
1.4422
4.63
1.5326
503
16154
543
I69I9
1.4446
464
1.5347
5.04
.6,74
544
16938
1.4469
, 465
1.5369
SOS
.6194
S45
16956
14493
4.66
■■5390
5.06
16214
546
16974
1.4516
4.67
1.5412
5.07
.6233
547
16993
1.4540
4.68
1.S433
5.08
.6253
S48
17011
1.4563
4.69
1.5454
5.09
16273
549
17029
1.4586
4.70
1.5476
5.1"
16292
550
17047
1.4609
4.71
1.5497
511
.6312
551
17066
14633
472
1.5518
5.12
.6332
552
17084
1.4656
473
1 5539
513
1635.
S53
I7I02
1.4679
474
1.5560
5.14
1637.
554
17.20
1.4702
475
1.55s.
5.15
16390
555
.7.38
14725
4.76
1.5602
5.16
16409
5s6
17156
'.4748
4.77
1.5623
517
,6429
557
17174
1.4770
4.78
1.5644
5.18
16448
5.5s
17192
14793
479
1.5665
5.19
,6467
559
17210
1.48.6
4.80
1.5686
5.20
.6487
560
17228
HYPERBOLIC LOGARITHMS OF NUMBERS.
63
i Number. Logarithm.
Number.
6.01
6.02
6.03
6.04
6.05
Logarithm.
Number.
Logarithm.
Number.
Logarithm.
5.61 1.7246
5.62 1.7263
5.63 1. 7281
•5.64 1.7299
565 I 1. 7317
1
1.7934
I795I
1.7967
1.7984
1. 800 1
6.41
6.42
6.43
6.44
6.45
1.8579
1.8594
I.8610
1.8625
1. 8641
6.81
6.82
6.83
6.84
6.85
1.9184
I.9199
I.9213
1.9228
1.9242
566
5.67
5.68
569
570
17334
17352
1.7370
17387
1.7405
6.06
6.07
6.08
6.09
6.10
1. 8017
1.8034
1.8050
1.8066
1.8083
i 6.46
6.47
6.48
6.49
6.50
1.8656
1.8672
1.8687
1.8703
I.8718
6.86
6.87
6.88
6.89
6.90
1.9257
1.9272
1.9286
I.93OI
1.9315
571
572
573
574
1 575
1
1.7422
1.7440
1.7457
1.7475
1.7492
6.U
6.12
6.13
6.14
6.15
1
1.8099
1.8116
1.8132
1.8148
1.8165
6.51
6.52
6.53
' 6.54
: 6.55
1.8733
1.8749
1.8764
1.8779
1.8795
6.91
6.92
6.93
i 6.94
6.95
1.9330
1.9344
1.9359
19373
19387
576
577
578
579
580
1.7509
1.7527
1.7544
I. 7561
1.7579
6.16
6.17
6.18
6.19
6.20
i.8i8i
1.8197
1.8213
1.8229
1.8245
6.56
6.57
6.58
6.59
• 6.60
1
1.8810
1.8825
1.8840
1.8856
1.8871
6.96
! 6.97
6.98
6.99
7.00
1.9402
1. 9416
L 1.9430
1.9445
1.9459
581
58*
583
584
585
1.7596
I. 7613
1.7630
1.7647
1 . 7664
6.21
6.22
6.23
6.24
6.25
1.8262
1.8278
1.8294
1.8310
1.8326
6.61
6.62
6.63
6.64
6.65
1.8886
1. 8901
1. 8916
1.8931
1.8946
7.01
7.02
7.03
7.04
7.05
19473
1.9488
1.9502
1.9516
1.9530
586
587
5.88
589
5.90
1. 7681
1.7699
1. 7716
17733
1.7750
6.26
6.27
6.28
6.29
6.30
1
1.8342
1.8358
18374
1.8390
1.8405
6.66
6.67
6.68
6.69
6.70
1. 8961
1.8976
1.8991
1.9006
1. 9021
7.06
7.07
7.08
7.09
7.10
1.9544
1.9559
1.9573
1.9587
1. 9601
591
592
593
594
595
1.7766
1.7783
1.7800
1. 7817
1.7834 1
1
6.31
6.32
6.33
6.34
6.35
1. 8421
1.8437
1.8453
1.8469
1.8485
' 6.71
6.72
6.73
6.74
6.75
1.9036
1.9051
1.9066
1. 9081
1.9095
7.II
7.12
7.13
7.14
7.15
1.9615
1.9629
1.9643
1.9657
1.9671
596
597
5.98
599
. 6.00
1
•
1.7851
1.7867
1.7884
1.7901
1. 7918
6.36
6.37
6.38
6.39
6.40
1.8500
1.8516
1.8532
1.8547
1.8563
6.76
6.77
. 6.78
1 6.79
6.80
1.9110
1.9125
1. 9140
1.9155
1. 9169
7.16
7.17
7.18
7.19
7.20
1.9685
1.9699
1.9713
1.9727
1. 9741
MATHEMATICAL TABLES.
I^,g«i*m.
Number.
^.^.
Number.
Logarilhn..
»„,»,.
L.«uithin.
I97SS
1.9769
..9782
7,61
7.62
763
764
765
2.0295
2.0308
2.0321
Z0334
20347
8.01
8.02
8.03
8.04
8.05
2.0807
2.0819
2.0832
2.0844
2.0857
8.41
8.42
8.43
8.44
845
2.1294
2.1330
2.1342
1.9824
1.9838
1.9851
1.986s
1.9879
7.66
7.67
7.68
7.69
7.70
2.0360
20373
2.0386
2.0399
2.0412
8.06
8.07
8.08
8.09
8.10
2.0869
2.0882
2.0894
2.0906
2.0919
S.46
8.47
8.48
If.
21353
2.1365
2.1377
2.1389
2.1401
■.9892
1.9906
1.9920
1.9933
19947
7.71
7.72
773
7.74
775
20425
20438
2.0451
2.0464
2.0477
8.1 1
8.12
8.13
8.14
8.15
2.0931
2.0943
2.0980
8.51
8.52
8.53
8.54
8.SS
2.1412
2.1424
2.1436
2.1448
2.1459
,.996,
■9974
..9988
2.0001
2.0015
7.76
777
778
779
7.80
2.0490
2.0503
2.0516
2.0528
2.0541
8.16
8.17
8.18
8.19
8.30
2.0992
2.I00S
2.1017
2.1029
2.IO4I
8.56
8.57
8.58
8.59
8.60
2.1471
2.1483
2.1494
2.1506
2.1518
2.0028
2.0042
2005 S
2.0069
2.0082
7.81
7.82
783
784
78s
2.0554
2.0567
2.0580
2.0605
8.21
8.22
8.23
8.24
8.25
2.1054
2,1066
2.1078
2. 1090
8.61
8.62
8.63
8.64
8.65
2.1529
2.1541
2.1552
2.1564
2.15,6
2.0096
2.0109
2.0149
7.86
787
7.88
7.89
7.90
2.0618
2.0631
2.0643
2.0656
2.0669
8.26
8.27'
8.28
8.29
8.30
2!lI26
2.1138
2. I 150
2.I163
8.66
8.67
8.68
8.69
8.70
2.1587'
2.1599
2.1610
2.1622
2.1633
2.0162
2.0176
2'0202
2.0215
7.91
7.92
793
794
795
2.0681
2.0694
2.0707
2.0719
2.0732
8.31
8.32
8.33
8.34
83S
2.H99
2.12II
2.1223
8.71
8.72
8.73
8.74
8.75
2.1645
2.1656
2.1668
2.1679
2.169,
2.0229
2.0242
202SS
2.0268
2.0281
7.96
7.97
7.98
799
8.00
2.0744
2.0757
2.0769
2.0782
20794
8.36
8.37
8.38
8.39
8.40
2.1235
2.1247
2.1258
2.1270
2.1282
8.,6
877
8.78
8.79
8.80
2.1702
2.1713
2.1725
2.1,36
2.1748
HYPERBOLIC LOGARITHMS OF NUMBERS.
65
t
Nanber.
Logarithm.
Number.
Logarithm.
Number.
Logarithm.
Number.
Logarithm.
8.81
2.1759
9.II
2.2094
9.41
2.2418
9.71
2.2732
8.82
2.1770
9.12
2.2105
9.42
2.2428
9.72
2.2742
8.83
2.1782
913
2.2116
9.43
2.2439
9.73
2.2752
8.84
2.1793
9.14
2.2127
944
2.2450
9.74
2.2762
8.85
2.1804
915
2.2138
9.45
2.2460
9.75
2.2773
8.86
2.1815
9.16
2.2148
9.46
2.2471
9.76
2.2783
8.87
2.1827
9.17
2.2159
9.47
2.2481
977
2.2793
8.88
2.1838
9.18
•2.2170
9.48
2.2492
9.78
2.2803
8.89
2.1849
9.19
2.2181
9.49
2.2502
9.79
2.2814
8.90
2.1861
9.20
2.219^
9.50
2.2513
9.80
2.2824
8.91
2.1872
9.21
2.2203
9.51
2.2523
9.81
2.2834
8.92
2.1883
9.22
2.2214
9.52
2.2534
9.82
2.2844
893
2. 1 894
923
2.2225
9.53
2.2544
9.83
2.2854
8.94
2. 1 905
9.24
2.2235
9.54
2.2555
9.84
2.2865
8.95
2.1917
9.25
2.2246
, 9.55
2.2565
9.85
2.2875
8.96
2.1928
9.26
2.2257
9.56
2.2576
9.86
2.2885
8.97
2.1939
9.27
2.2268
9.57
2.2586
9.87
2.2895
8.98
2.1950
9.28
2.2279
9.58
2.2597
9.88
2.2905
8.99
2.1961
9.29
2.2289
959
2.2607
9.89
2.2915
9.00
2.1972
930
2.2300
' 9.60
2.2618
9.90
2.2925
9.01
2.1983
931
2.23II
9.61
2.2628
9.91
2.2935
9.02
2.1994
932
2.2322
9.62
2.2638
9.92
2.2946
903
2.2006
933
2.2332
9.63
2.2649
993
2.2956
9.04
2.2017
934
2.2343
9.64
2.2659
9.94
2.2966
9.05
2.2028
935
2.2354
9.65
2.2670
995
2.2976
9.06
2.2039
936
2.2364
i 9.66
2.2680
9.96
2.2986
9.07
2.2050
937
2.2375
i 967
2.2690
997
2.2996
9.08
2.2061
938
2.2386
9.68
2.2701
9.98
2.3006
: 9.09
2.2072
939
2.2396
9.69
2.271I
9.99
2.3016
9.10
1
2.2083
9.40
2.2407
9.70
2.2721
10.00
2.3026
10.25
2.3279
12.75
2.5455
15.50
2.7408
1
j 21.0
3.0445
10.50
2.3513
13.00
2.5649
16.0
2.7726
22.0
3.O9II
10.75
2.3749
13.25
2.5840
16.5
2.8034
23.0
3.1355
11.00
2.3979
1350
2.6027
17.0
2.8332
24.0
3.I781
11.25
2.4201
13.75
2.62II
17.5
2.8621
25.0
3.2189
11.50
2.4430
14.00
2.6391
18.0
2.8904
26.0
3.2581
"75
2.4636
14.25
2.6567
18.5
2.9173
: 27.0
3.2958
12.00
2.4849
14.50
2.6740
19.0
2.9444
28.0
3.3322
12.25
2.5052
14.75
2.6913
19.5
2.9703
29.0
33673
12.50
2.5262
15.00
2.7081
20.0
2.9957
30.0
3.4012
MATHEMATICAL , TABLES.
[IL— NUMBERS, OR DIAMETERS OF CIRCLES, CIR
ENCES, AREAS, SQUARES, CUBES, SQUARE ROOTS,
BE ROOTS.
0.7854
3 14
7.07
1257
19.63
. 28.^7
38.48
. S0.26
63.61
78.54
9503
.113.09
13273
■•5393
17371
.aoi.o6
226.98
.254.46
28J.52
.3 14 15
34636
,380.13
41547
452.38
490,87
.53002
57^55
61575
660.52
.706.85
75476
85529
.907.92
962.11
:oi7.87
.075.21
13411
.. 256
289 I
1.331
,.1,728
2,197 !
.. 2,744
3375
,.4,096
4913
. 5,832
6,859
■9:261
15,62s
17,576
19,683
29,791
3^,768
35,937
39,3°4
42,875
46,656
50,653
54,872
59,3 '9
64,000
68,921
74,088
3162
3316
3464
3.605
5741
3.872
4.000
4123
4.242
4358
4472
4582
4795
4.898
5.000
5099
5196
5291
5385
S477
5567
5.656
5744
5830
5.916
6.000
6.082
6.164
6.244
6.324
6.403
6.480
NUMBERS, OR DIAMETERS OF CIRCLES, &c.
67
i?
Nttjnber,
Circum
CtroUar
Sauaie.
Cube.
Square
.Cube
Diameter.
ference.
Area.
^^«4 t*a«a^««
Root.
Root
43
135oS
1452.20
1,849
79,507
6.557
3.503
44
138.23
... 1520.52
... 1,936
85,184
6.633
3.530
45
141.37
1590.43
2,025
91,125
6.708
3.556
46
144.51
... 1661.90
... 2,116
97,336
6.782
3.583
47
147.65
1734.94
2,209
103,823
6.855
3.608
48
15079
... 1809.55
... 2,304
110,592
6.928
3634
49
153.93
1885.74
2,401
117,649
7.000
3.659
50
157.08
... 1963.49
... 2,500
125,000
7.071
3.684
51
160.22
2042.82
2,6oi
132,651
7.I4I
3.708
52
163.36
... 2123.71
..: 2,704
140,608
7. 211
3.732
53
166.50
2206.18
2,809
148,877
7.280
3.756
54
169.64
... 2290.21
... 2,916
157,464
7.348
3.779
55
172.78
2375.82
3»025
166,375
7.416
3.802
56
17592
... 2463.09
... 3*136
I75»6i6
7.483
3.825
57
179.07
2551.75
3i249
185,193
7549
3.848
58
182.21
... 2642.08
... 3*364
i95,"2
7.61S
3.870
59
18535
273397
3,481
205,379
7.681
3.892
60
188.49
... 2827.43
... 3,600
216,000
7.745
3.914
61
19^63
2922.46
3,721
226,981
7.810
3.936
62
194.77
... 3019.07
... 3,844
238,328
7.874
3.957
63
197.92
3117.24
3,969
250,047
7.937
3.979
64
201.06
... 3216.99
... 4^096
262,144
8.000
4.000
65
204.20
3318.30
4,225
274,625
8.062
4.020
66
207.34
... 3421.18
... 4,356
287,496
8.124
4.041
67
210.48
3525.65
4,489
300,763
8.185
4.061
68
213.62
... 3631.68
... 4,624
314,432
8.246
4.081
69
216.77
3739.28
4,761
328,509
8.306
4.IOI
70
219.91
... 3848.45
... 4,900
343,000
8.366
4. 12 I
71
223.05
3959.19
5,041
357,911
8.426
4.140
72
226.19
... 4071.50
... 5,184
373,248
8.485
4.160
73
22g.Z2»
4185.38
5,329
389,017
^.544
4.179
. 74
^S^Al
... 4300.84
... 5,476
405,224
8.602
4.198
1
75
235.61
4417.86
5,625
421,875
8.660
4.217
76
238.76
... 4536.45
••• 5,776
438,976
8.717
4.235
77
241.90
4656.62
5,929
456,533
8.744
4.254
78 1
24504
... 4778.36
... 6,084
474,552
8.831
4.272
79 '
248.18
4901.66
6,241
493,039
8.888
4.290
1 W ^
80
25132
... 5026.54
... 6,400
512,000
8.944
4.308
81 ' 25446
5153.00
6,561
531,441
9.000
4.326
82 257.61
... 5281.01
... 6,724
551,368
9.055
4.344
83 260.75
5410.59
6,889
571,787
9. 1 10
4.362
84
^es^9
... 5541.77
... 7,056
592,704
9.165
4.379
85
%6
267.03
5674.50
7,225
614,125
9.219
4.396
27017
... 5808.80
... 7,396
636,056
9.273
4.414
87
2733^
5944.67
7,569
658,503
9.327
4.431
88
276.46
... 6082.11
... 7,744
681,472
9.380
4.447
89
90
279.60
6221.13
7,921
704,969
9.433
4.461
282.74
... 6361.72
... 8,100
729,000
9.486
4.481
" ill
68
MATHEMATICAL TABLES.
Number,
1
or
Circum
Circular
Sfluare.
Cube.
Square
Cube
Diameter.
ference.
Area.
^^%»A#^»«
Root.
Root.
91
285.88
6503.87
8,281
753*571
9.539
4.497
92
289.02
... 6647.61
... 8,464
778,688
9591
4.514
93
292.16
6792.90
8,649
804,357
9.643
4.530
94
29531
... 6939.78
... 8,836
830,584
9.695
4.546
95
298.45
7088.21
9>o25
857*375
9.746
4.562
96
30159
... 7238.23
... 9,216
884,736
9.797
4.578
97
304.73
7389.81
9,409
912,673
9.848
4.594
98
307.87
... 7542.96
... 9,604
941,192
9.899
4.610
99
3II.OI
7697.68
9,801
970,299
9.949
4.626
100
314.15
. 7853.97
...10,000
... 1,000,000
10.000
4.641
lOI
317.30
8011.86
10,201
1*030,301
10.049
4.657
102
320.41
... 8171.30
...10,404
... 1,061,208
10.099
4.672
103
323.58
8332.30
10,609
1,092,727
10.148
4.687
104
326.72
... 8494.88
...10,816
.,.. 1,124,864
10.198
4.702
105
329.86
8659.03
11,025
1*157,625
10.246
4.717
106
333.00
... 8824.75
!... 11,236
... 1,191,016
10.295
4.732
107
336.15
8992.04
11,449
1,225,043
10.344
4.747
108
339.29
... 9160.90
...11,664
... 1,259,712
10.392
4.762
109
342.43
9331.33
11,881
1,295,029
10.440
4.776
no
34557
 9503.34
...12,100
■•• 1*331,000
10.488
4.791
III
348.71
9676.91
12,321
1,367.631
10.535
4.805
112
351.85
... 9852.05
...12,544
... 1,404,928
10.583
4.820
113
355.01
10028.77
12,769
1,442,897
10.630
4.834
114
358.14
...10207.05
...12,996
••• 1,481,544
10.677
4.848
115
361.28
10386.91
13*225
1,520,875
10.723
4.862
116
364.42
...10568.34
... 13*456
... 1,560,896
10.770
4.876
117
367.56
10751.34
13,689
1,601,613
10.816
4.890
118
370.70
...10935.90
...13,924
••• 1,643,032
10.862
4.904
119
373.81
III22.O4
14,161
1*685,159
10.908
4.918
120
376.99
...11309.76
...14,400
... 1,728,000
10.954
4.932
121
380.1^
11499.04
14,641
1,771,561
11.000
4.946
122
383.27
...11689.89
...14,884
... 1,815,848
11.045
4.959
123
386.41
I1882.3I
15*129
1,860,867
11.090
4.973
124
389.55
...12076.31
...15*376
... 1,906,624
11.135
4.986
125
392.70
12271.87
15*625
1*953,125
II. 180
5.000
126
395.84
...12469.01
...15*876
... 2,000,376
11.224
5013
127
398.98
12667.71
16,129
2,048,383
11.269
5.026
128
402.12
...12867.99
...16,384
... 2,097,152
11.313
5.039
129
405.26
13069.84
16,641
2,146,689
11.357
5.052
130
408.10
...13273.26
...16,900
... 2,197,000
1 1. 401
5.065
131
411.54
13478.24
17,161
2,248,091
11.445
5.078
132
414.69
...13694.80
...17,424
... 2,299,968
11.489
5.091
133
417.83
13892.94
17,689
2,352,637
".532
5.104
134
420.97
...14102.64
...17,956
... 2,406,104
.11.575
5."7
135
424.11
I4313.9I
18,225
2,460,375
II. 618
5.129
136
427.25
...14526.75
...18,496
... 2,515,456
II. 661
5.142
137
430.39
I474I.I7
18,769
2*571,353
11.704
5155
138
1
433.54
••.14957.15
...19,044
... 2,620,872
11.747
5.167
NUMBERS, OR DIAMETERS OF CIRCLES, ftc.
69
Nanaber,
nr
Grcnm
Circular
Square.
Cube.
Square
Cube
EKaxneter.
f«rence. .
Area.
Root.
Root.
139
436.68
I5174.7I
19,321
2,685,619
11.789
5.180
140
439.82
•••15393.84
...19,600
... 2,744,000
11.832
5192
141
442.96
1561453
19,881
2,803,221
11.874
5.204
142
446.10
...15836.80
...20,164
... 2,863,288
11.916
5217
143
449.24
16060.64
20,449
2,924,207
11.958
5.229
144
452.39
...16286.05
...20,736
••• 2,985,984
12.000
5241
145
455.53
16513.03
21,025
3,048,625
12.041
5253
146
45».67
...16741.58
...21,316
... 3,112,136
12.083
5.265
147
461.81
16971.70
21,609
3,176,523
12.124
5277
148
46495
...17203.40
...21,904
••• 3,241,792
12.165
5.289
149
468.09
17436.66
22,201
3,307,949
12.206
5.301
150
471.24
...17671.50
...22,500
••• 3,375,000
12.247
5.313
151
474.3S
17907.90
22,8oi
3,442,951
12.288
5.325
152
477.52
...18145.88
...23,104
... 3,511,808
12.328
5336
153
480.66
18385.42
23,409
3,581,577
12.369
5.348
154
483.80
...18626.54
...23,716
... 3,652,264
12.409
5.360
155 ,
486.94
18869.23
24,025
3,723,875
12.449
5.371
156
490.08
...19113.49
••.24,336
... 3,796,416
12.489
5.383
157
493.23
1935932
24,649
3,869,893
12.529
5.394
158
496.37
...19606.72
...24,964
... 3,944,312
12.569
5.406
159
499.51
19855.69
25,281
4,019,679
12.609
5.417
160
502.65
...20106.24
...25,600
... 4,096,000
12.649
5.428
161
50579
20358.35
25,921
4,173,281
12.688
5440
162
508.93
...20612.03
...26,244
... 4,251,528
12.727
5.451
163
512.08
20867.20
26,569
4,330,747
12.767
5.462
. 164
515.22
...21 124. 1 1
...26,896
... 4,410,944
12.806
5473
165
518.36
21382.51
27,225
4,492,125
12.845
S484
166
521.50
...21642.48
...27,556
••• 4,574,296
12.884
5.495
167
52464
21904.02
27,889
4,657,463
12.922
5506
168
527.78
...22167.12
...28,224
••• 4,741,632
^12.961
5517
169
53093
22431.80
28,561
4,826,809
13.000
5.528
170 1
53407
...22698.06
...28,900
••• 4,913,000
13.038
5.539
171
537.31
22965.88
29,241
5,000,211
13.076
5.550
172
540.35
•. 2323527
...29,584
... 5,088,448
13.114
5.561
173
543.49
23506.23
29,929
5,177,717
13.152
5.572
174
546.03
...23778.77
...30,276
... 5,268,024
13.190
5.582
175
549. 78
. 24052.87
30,625
5,359,375
13.228
5.593
176 1
552.92
...24328.55
...30,976
••• 5,451,776
13.266
5.604
177
556.06
24605.79
31,329
5,545,233
13.304
5.614
178
55920
...24884.61
...31,684
 5,639,752
13341
5.625
179
56234
25165.00
32,041
5,735,339
13379
5.635
zSo
565.48
...25446,96
...32,400
... 5,832,000
13.416
5.646
idi
568.62
25730.48
32,761
5,929,741
13453
5.656
182
571.77
...26015.58
..•33,124
... 6,028,568
13.490
5.667
183 ;
574.91
26302.26
33,489
6,128,487
13^527
5.677
184
573.05
...26590.50
...33,856
... 6,229,504
13564
5.687
. 1S5
581.19
26880.31
34,225
6,331,625
13.601
5.698
186
584.33
...27171.69
••34,596
... 6,434,856
13.638
5.708
70
MATHEMATICAL TABLES.
Number,
or
Circum
Cirailar
Square.
Cube.
Square
Cube
Diameter.
ference.
Area.
Root.
Root.
187
537.47
27464.65
34,969
6,539,203.
13.674
5.718
188
590.62
..•2775917
•35>344
... 6,644,672
I3.71I
5.728
189
59376
28055.27
35,721
6,751,269
13747
5.738
190
596.90
...28352.94
...36,100
... 6,859,000
13.784
5.748
191
600.04
28652.17
36,481
6,967,871
13.820
5.758
192
603.18
...28952.98
...36,864
... 7,077,888
13.856
5.768
193
606.32
29255.36
37,249
7,189,057
13.892
5.778
194
* 609.47
29559.31
...37,636
... 7,301,384
13.928
5.788
195
612.61
29864.83
38,025
7,414,875
13.964
5.798
196
615.75
...30171.92
...38,416
 7,529,536
14.000
5.808
197
618.89
30480.60
38,809
7,645,373
14.035
5.818
198
622.03
...30790.82
...39,204
... 7,762,392
14.071
5.828
199
625.17
31102.52
39,601
7,880,599
14.106
5.838
200
628.32
...31416.00
...40,000
... 8,000,000
14.142
5.848
201
631.46
31730.94
40,401
8,120,601
14.177
5.857
202
634.60
...32047.46
...40,804
... 8,242,408
14.212
5.867
203
637.74
32365.54
41,209
8,365,427
14.247
5.877
204
640.88
...32685.20
...41,616
... 8,489,664
14.282
5886
205
644.02
33006.43
42,025
8,615,125
' 14.317
5.896
206
647.16
•33329.23
...42,436
... 8,741,816
14.352
5.905
207
65031
33653.60
42,849
8,869,743
14387
5915
208
65345
33979.54
...43,264
... 8,998,912
14.422
5924
209
656.59
34307.05
43,681
9,123,329
14.456
5.934
210
65973
...34636.14
...44,100
... 9,261,000
14.491
5.943
211
662.87
34966.79
44,521
9,393,931
14.525
5.953
212
666.01
...35299.01
••44,944
... 9,528,128
14.560
5.962
^13
669.16
35632.81
45,369
9,663,597
14.594
5.972
214
672.30
...35968.17
...45,796
... 9,800,344
14.628
5.981
215
67544
36305.11
46,225
9,938,375
14.662
5.990
216
678.5S
...36643.62
...46,656
...10,077,696
14.696
6.000
217
681.71
36983.70
47,089
10,218,313
14.730
6.009
218
684.86
••.37325.34
..•47,524
...10,360,232
14.764
6.018
219
688.01
37668.56 .
47,961
10,503,459
14.798
6.027
220
691.15
...38013.36
...48,400
...10,648,000
14.832
6.036
221
694.29
3835972
48,841
10,793,861
14.866
6.045
222
697.43
...38707.65
...49,284
...10,941,048
14.899
6.055
223
700.57
39037.51
49,729
11,089,567
14.933
6.064
224
70371
...39408.23
...50,176
...11,239,424
14.966
6.073
225
706.86
39760.87
50,625
11,390,625
15.000
6.082
226
710.00
...40115.09
...51,076
•",543,176
15033
6.091
227
713.14
40470.87
51,529
11,697,083
15.066
6.100
228
716.28
...40828.23
...51,984
••.11,852,352
15.099
6.109
229
719.42
41187.16
52,441
12,008,989
15132
6. 1 18
230
722.56
...41547.66
...52,900
...12,167,000
15165
6.126
231
725.70
41909.72
53,361
12,326,391
15.198
6.135
232
728.85
...42273.36
..53,824
...12,487,168
15.231
6.144
233
731.99
• 42638.58
54,289
12,649,337
15.264
6.^53
234
735.13
...43005.36
..54,756
...12,812,904
15.297
6.162
NUMBERS, OR DIAMETERS OF CIRCLES, &c.
71
NnTTiber,
or
Circum
Circular
Square.
Cube.
Square
Cube
Dumeter
ference.
• 1
Area.
Root.
Root.
235
i 738.27
4337371
55,225
12,977,875
15329
6.171
236
741.41
...4374363
...55,696
...13,144,256
15.362
6.179
237
74455
44115.11
56,169
13,312,053
15.394
6.188
23S
747.68
...44488.19
...56,644
...13,481,272
15.427
6.197
239
750.88
44862.83
57,121
13,651,919
15459
6.205
240
753.98
...45239.04
...57,600
...13,824,000
I549I
6.214
241
757.12
45616.81
58,081
13,997,521
15.524
6.223
242
760.26
...45996.16
...58,564
...14,172,488
15.556
6.231
243
763.40
46377.08
59,049
14,348,907
15.588
6.240
244
766.52
...46759.57
...59,536
...14,526,784
15.620
6.248
245
769.92
4714363
60,025
14,706,125
15^652
6.257
246
' 772.83
...47529.26
...60,516
...14,886,936
15.684
6.265
247
77597
47916.46
61,009
15,069,223
15.716
6.274
248
779.11
...48305.24
...61,504
...15,252,992
15.748
6.282
249
782.25
48695.58
62,001
15,438,249
15.779
6.291
250
785.40
...49087.50
...62,500
...15,625,000
I5.81I
6.299
251
788.54
49480.98
63,001
15,813,251
15.842
6.307
252
791.68
...49876.04
...63,504
...16,003,008
15.874
6.316
253
794.82
50272.66
64,009
16,194,277
15.905
6.324
254
797.96
...50670.86
...64,516
...16,387,064
15.937
6.333
255
801.10
51070.63
65,025
16,581,375
15.968
6.341
256
804.24
...5147196
...65,536
...16,777,216
16.000
6.349
257
807.39
51874.88
66,049
16,974,593
16.031
6.357
258
810.53
...52279.36
...66,564
••.17,173,512
16.062
6.366
259
813.67
52685.41
67,081
17,373,979
16.093
6.374
2€o
816.81
...53093.04
...67,600
•..17,576,000
16.124
6.382
, 261
819.95
53502.23
68,121
17,779,581
16.155
6.390
j 262
823.09
...53912.99
...68,644
...17,984,728
16.186
6.398
263
826.24
54325.33
69,169
18,191,447
16.217
6.406
264
829.38
54739.23
...69,696
•••18,399,744
16.248
6.415
265
832.52
55154.71
70,225
18,609,625
f6.278
6.423
266
835.66
...55571.76
...70,756
...18,821,096
16.309
6.431
267
838.80
55990.38
71,289
19,034,163
16.340
6.439
268
841.94
...56410.56
...71,824
...19,248,832
16.370
6.447
269
84509
56832.32
72,361
19,465,109
16.401
6.455
270
848.23
...57255.66
...72,900
...19,683,000
16.431
6.463
271
851.37
57680.56
73,441
19,902,511
16.462
6.471
272
85451
...58107.03
...73,984
...20,123,648
16.492
6.479
. ^^^
857.65
58535.07
74,529
20,346,417
16.522
6.487
1 274
860.79
...58964.69
...75,076
...20,570,824
16.552
6.495
275
863.94
59393.87
75,625
20,796,875
16.583
6.502
276 ;
867.08
...59828.63
...76,176
...21,024,576
16.613
6.510
277
870.22
60262.95
76,729
21,253,933
16.643
6.518
278
873.36
...60698.85
...77,284
...21,484,952
16.673
6.526
279
876.50
61136.32
77,841
21,717,639
16.703
6.534
280
879.64
...61573.36
...78,400
...21,952,000
16.733
6.542
281
882.78
62015.96
78,961
22,188,041
16.763
6.549
282
885.93
...62458.14
...79,524
...22,425,768
16.792
6.557
/
/
/
\
72
MATHEMATICAL TABLES.
Number.
A « «• AAA l^^^s J
or
Circum
Circular
Square.
Cube.
Square
Cube
Diameter.
ference.
Area.
Root.
Root.
283
889.07
62901.90
80,089
22,665,187
16.822
6.565
284
892.21
...63347.22
...80,656
...22,906,304
16.852
6.573
285
89535
63794.11
81,225
23,149,125
16.881
6.580
286
898.49
...64242.57
...81,796
•••23,393,656
16.9II
6.588
287
901.63
64692.61
82,369
23,639,903
16.941
6.596
288
904.78
...65144.21
...82,944
...23,887,872
16.970
6.603
289
907.92
6559739
83,521
24,137,569
17.000
6.611
290
911.06
...66052.14
...84,100
...24,389,000
17.029
6.619
291
914.20
66508.45
84,681
24,642,171
17.059
6.627
292
91734
...66966.34
...85,264
...24,897,088
17.088
6.634
293
920.48
67425.80
85,849
25*153,757
I7.II7
6.642
294
923.63
...67886.83
...86,436
...25,412,184
17.146
6.649
295
926.77
68349.43
87,025
25*672,375
17.176
6.657
296
929.91
...68813.60
...87,616
...25,934,336
17.205
6.664
297
93305
69279.34
88,209
26,198,073
17.234
6.672
298
936.19
...69746.66
...88,804
...26,463,592
17.263
6.679
299
93933
. 70215.54
89,401
26,730,899
17.292
6.687
300
942.48
...70686.00
...90,000
...27,000,000
17.320
6.694
301
945.62
71158.02
90,601
27,270,901
17.349
6.702
302
948.76
...71631.62
...91,204
...27,543,608
17.378
6.709
303
95190
72106.78
91,809
27,818,127
17.407
6.717
304
95504
••72583.52
...92,416
...28,094,464
17.436
6.724
305
958.18
73061.83
93,025
28,372,625
17.464
6.731
306
961.32
..•73541.71
•93,636
...28,652,616
17.493
6.739
307
964.47
74023.16
94,249
28,934,443
^7.521
6.746
308
967.61
...74506.18
...94,864
...29,218,112
17.549
6.753
309
970.75
74990.77
95,481
29,503,629
17578
6.761
310
97389
...75476.94
...96,100
...29,791,000
17.607
6.768
311
97703
75964.67
96,721
30,080,231
17.635
6.775
312
980. 1 7
...76453.93
...97,344
.. 30,371*328
17.663
6.782
313
983.32
7694485
97,969
30,664,297
17.692
6.789
314
986.45
...77437.29
...98,596
...30,959,144
17.720
6.797
315
989.60
77931.31
99,225
31,255,875
17.748
6.804
316
992.74
...78426.89
...99,856
.. .31*554,496
17.776
6.81 1
317
99588
78924.06
100,489
31*855,013
17.804
6.818
318
999.02
...79422.78
101,124
•..32,157,432
17.832
6.826
319
1002.17
79923.08
101,761
32,461,759
17.860
6.833
320
1005.31
...80424.96
102,400
...32,768,000
17.888
6.839
321
1008.45
80928.40
103,041
33,076,161
17.916
6.847
322
1011.59
...81433.41
103,684
...33*386,248
17944
6.854
323
1014.73
81939.99
104,329
33,698,267
17.97^2
6.861
324
1017.47
...82448.15
104,976
...34,012,224
18.000
6.868
325
1021.02
82957.87
105,625
34,328,125
18.028
6.875
326
1024.16
...83469.17
106,276
..•34,645*976
18.055
6.882
327
1027.30
83982.60
106,929
34,965*783
18.083
6.889
328
1030.44
...84496.47
107,584
...35*287,552
18.IH
6.896
329
1033.58
85012.48
108,241
35,611,289
18.138
6.903
330
1036.72
...85530.06
108,900
•.35,937*000
18.166
6.910
NUMBERS, OR DIAMETERS OF CIRCLES, Ac.
73
Nomber,
or
Grcum
Circular
Square.
Cube.
Square
Cube
Diameter.
ference.
Area.
Root.
Root.
331
1039.86
86049.20
109,561
36,264,691
18.193
6.917
332
1043.01
...86569.92
110,224
...36,594,368
18.221
6.924
333
1046.15
87092.22
110,889
36,926,037
18.248
6.931
334
1049.29
...87616.08
111,556
..•37,259,704
18.276
6.938
335
1052.43
88141.51
112,225
37,595,375
18.303
6.945
336
1055.57
...88668.51
112,896
...37^33,056
18.330
6.952
337
1058.71
89197.09
113,569
38,272,753
18.357
6.959
338
1061.86
...89727.23
114,244
...38,614,472
18.385
6.966
339
1065.02
90258.9s
114,921
38,958,219
18.412
6.973
340
1068.14
...90792.24
115,600
...39,304,000
18.439
6.979
341
1071.28
91327.09
116,281
39,651,821
18.466
6.986
342
1074.27
■..91863.52
116,964
...40,001,688
18.493
6.993
343
1077.56
92401.15
117,649
40,353,607
18.520
7.000
344
1080.71
...92941.09
118,336
...40,707,584
18.547
7.007
345
1083.85
93482.23
119,025
41,063,625
18.574
7.014
346
1086.99
...94024.94
119,716
...41,421,736
18.601
7.020
347
1090.35
94569.22
120,409
41,781,923
18.628
7.027
348
1093.07
...95115.08
121,104
...42.144,192
18.655
7.034
349
1096.41
95662.50
I2I,8oi
42,508,549
18.681
7.040
350
1099.56
...96211.50
122,500
...42,875,000
18.708
7.047
351
1102.70
96762.06
123,201
43,243,551
18.735
7.054
352
1105.84
...97314.20
123,904
...43,614,208
18.762
7.061
353
1 108.98
97867.90
124,609
43,986,977
18.788
7.067
354
III2.62
...98423.18
125,316
...44,361,864
18.815
7.074
355
III5.26
98980.03
126,025
44,738,875
18.842
7.081
356
1 118.40
..•99538.45
126,736
...45,118,016
18.868
7.087
357
II2I.55
100098.43
127,449
45,499,293
18.894
7.094
358
1124.69
100660.00
128,164
...45,882,712
18.921
7.101
359
1127.83
101223.13
128,881
46,268,279
18.947
7.107
360
1130.97
101787.84
129,600
...46,656,000
18.974
7.114
361
II34.II
102354. H
130,321
47,045,881
19.000
7.120
362
, "37.25
102921.95
131,044
.••47,437,928
19.026
7.127
3^3
1140.40
103491.31
131,769
47,832,147
19.052
7.133
364
1143.54
104062.35
132,496
...48,228,544
19.079
7.140
365
1146.68
104634.91
133,225
48,627,125
19.105
7.146
366
1 149.82
105209.04
133,956
...49,027,896
19.131
7.153
367
1152.96
10578474
134,689
49,430,863
19.157
7.159
368
1 156.10
106362.00
135,424
•..49,836,032
19.183
7.166
369
1159.25
106940.84
136,161
50,243,409
19.209
7.172
370
1162.39
107521.26
136,900
..50,653,000
19.235
7.179
371
1165.53
108103.22
137,641
51,064,811
19.261
7.185
372
1168.67
108686.79
138,384
...51,478,848
19.287
7.192
373
II7I.81
IO9271.91
139,129
51,895,117
19.313
7.198
t
374
1 17495
109858.62
139,876
...52,313,624
19.339
7.205
375
II78.IO
110446.87
140,625
52,734,375
19.365
7. 211
376
1 181.24
IIIO36.71
141,376
...53,157,376
19391
7.218
377
1 18438
III628.II
142,129
53,582,633
19.416
7.224
378
1187.52
II222I.O9
142,884
...54,010,152
19.442
7230
74
MATHEMATICAL TABLES.
i Number,
or
Circumb
Circular
Square.
Cube.
Square (
iTube
Diameter.
ference.
Arcx
Root. I
looL
379
1190.66
II2815.64
143,641
54,439,939
19.468 7.
237
380
1193.80
II34II.76
144,400
...54,872,000
19.493 7
•243
381
1196.94
114009.46
145,161
55,306,341
19.519 7.
249
382
1200.09
114608.70
145*924
•..55,742,968
19545 7'
256
383
1203.23
115209.54
146,689
56,181,887
19.570 7.
,262
384
1206.37
115811*94
147,456
...56,623,104
19.596 7.
268
385
1209.51
I16415.9I
148,225
57,066,625
19.621 7.
'275
386
1212.65
117021.45
148,996
...57,512,456
19.647 7.
.281
387
1215.79
117628.57
149,769
57,960,603
19.672 7
.287
388
1218.94
118237.25
150,544
...58,411,072
19.698 7.
.294
389
1222.08
I18846.51
151,321
58,863,869
19.723 7.
299
390
1225.22
119453.94
152,100
••59,319,000
19.748 7
■306
391
1228.36
120072.73
152,881
59,776,471
19.774 7.
312
392
1231.50
120687.70
153,664
...60,236,288
19.799 7<
319
393
1234.64
121304.24
154,449
60,698,457
19.824 7,
.325
394
1237.79
121922.43
155,236
...61,162,984
19.849 7.
331
395
1240.93
122542.03
156,025
61,629,875
19875 7
337
396
1244.07
123163.28
156,816
...62,099,136
19.899 7
343
397
1247.21
123786.10
157,609
62,570,773
19.925 7
■349
398
1250.35
I24412.IO
158,404
•..63,044,792
19.949 7
.356
399
1253.49
125036.46
159,201
63,521,199
19.975 7
362
400
1256.64
125664.00
160,000
...64,000,000
20.000 7
■368
401
1259.78
126293.10
160,801
64,481,201
20.025 7.
.374
402
1262.92
126923.88
161,604
...64,964,808
20.049 7
.380
403
1266.06
127556.02
162,409
65,450,827
20.075 7.
.386
404
1269.20
128189.84
163,216
••65,939,264
20.099 7.
•392
405
1272.34
128825.23
164,025
66,430,125
20.125 7
•399
406
1275.48
129462.19
164,836
...66,923,416
20.149 7.
405
407
1278.63
I30IOO.71
165,649
67,419,143
20.174 7.
.411
408
1281.77
130740.82
166,464
...67,911,312
20.199 7
.417
409
1284.91
131382.49
167,281
68,417,929
1 20.224 7
.422
410
1288.05
132025.74
168,100
...68,921,000
20.248 7.
.429
411
I29I.19
132670.55
168,921
69,426,531
20.273 7.
434
412
1294.32
133316.93
169,744
••69,934,528
20.298 7.
441
413
1297.48
133964.89
170,569
70,444,997
20.322 7
■447
414
1300.62
134614.41
171,396
■•70,957,944
20.347 7
■453
415
130376
135265.51
172,225
71,473,375
20.371 7.
■459
416
1306.90
I35918.18
173,056
...71,991,296
20.396 7.
465
417
1310.04
136572.42
173,889
72,511,713
20.421 7.
.471
418
I313.18
137228.22
174,724
• 73,034,632
20.445 7
477
419
1316.32
137885.69
175,561
73,560,059
20.469 7.
.483
420
131947
138544.56
176,400
...74,088,000
20.494 7.
489
421
1322.61
139205.08
177,241
74,618,461
20.518 7
•495
422
1325.75
139867.17
178,084
...75,151,448
20.543 7
.501
423
1328.89
140530.83
178,929
75,686,967
20.567 7
■507
424
1332.03
I41196.07
179,776
...76,225,024
20.591 7.
•513
425
133518
141862.87
180,625
76,765,625
20.615 7
.518
426
1338.32
142531.25
181,476
•.■77,308,776
\ 20.639 7
■524
NUMBERS, OR DIAMETERS OF CIRCLES, 4c.
75
Nnmber,
or
Grcum
Circular
S<]uare.
Cube.
Square
Cube
Diameter.
ference.
Area.
Root.
Root.
427
1341.46
I432OI.I9
182,329
77,854,483
20.664
7.530
428
1344.60
143872.71
183,184
...78,402,752
20.688
7.536
429
1347.74
144545.80
184,041
78,953,589
20.712
7.542
430
1550.88
145220.46
184,900
...79,507,000
20.736
7.548
431
1354.02
145696.68
185,761
80,062,991
20.760
7.554
432
1357.17
14657448
186,624
...80,621,568
20.785
7559
433
1360.33
147253.85
187,489
81,182,737
20.809
7.565
434
1363.45
147934.80
188,356
...81,746,504
20.833
7.571
435
1366.59
I48617.3I
189,225
. 82,312,875
20.857
7.577
436
1369.73
149301.39
190,096
...82,881,856
20.881
7.583
437
1372.87
149987.05
190,969
83,453,453
20.904
7.588
438
1376.02
150674.27
191,844
...84,027,672
20.928
7.594
439
1379.16
151362.87
192,721
84,604,519
20.952
7.600
440
1382.30
152053.44
193,600
...85,184,000
20.976
7.606
441
1385.44
152745.37
194,481
85,766,121
21.000
7.612
442
1388.58
153438.88
195*364
.■.86,350,388
21.024
7.617
443
1391.72
154133.96
196,249
86,938,307
21.047
7.623
444
1394.87
154830.61
197,136
...87,528,384
21.071
7.629
445
1398.01
155528.83
198,025
88,121,125
21.095
7.635
446
I4OI.I5
156228.62
198,916
...88,716,536
21. 119
7.640
447
1404.29
156929.98
199,809
89,314,623
21.142
7.646
448
1407.43
157632.92
200,704
•••89,915,392
21.166
7.652
449
1410.57
158337.42
201,601
90,518,849
21.189
7.657
450
1413.72
159043.50
202,500
...91,125,000
21.213
7.663
451
1416.86
I5975I.I4
203,401
91,733,851
21.237
7.669
452
1420.00
160460.36
204,304
...92,345,408
21.260
7.674
453
1423.14
161171.14
205,209
92,959,677
21.284
7.680
454
1426.28
161883.50
206,106
•••93,576,664
21.307
7.686
455
1429.42
162597.43
207,025
94,196,375
21.331
7.691
456
1432.56
163312.93
207,936
...94,818,816
21.354
7.697
' 457
1435.71
164030.20
208,849
95,443,993
21.377
7703
45«
1438.85
164748.64
209,764
...96,071,912
21.401
7.708
459
1441.99
165468.85
210,681
96,702,579
21.424
7.714
460
144513
166190.64
211,600
• 97,336,000
21.447
7.719
, 461
1448.27
166913.99
212,521
97,972,181
21.471
7.725
462
I45I.4I
167638.91
213,444
...98,611,128
21.494
7.731
463
1454.56
168365.41
214,369
99,252,847
21.517
7.736
464
1457.70
169093.47
215,296
•••99,897,345
21.541
7.742
. 465
1460.84
169823.II
216,225
100,544,625
21.564
7.747
. 466
1463.98
17055432
2171I56
101,194,696
21.587
7.753
I 467
1467.12
I71287.IO
218,089
101,847,563
21.610
7.758
1 468
1470.26
172021.44
219,024
102,503,232
21.633
7.764
! 469
1473.41
172757.36
219,961
103,161,709
21.656
7.769
470
1476.55
173494.86
220,900
103,823,000
21.679
7^775
471
1479.69
174233.92
221,841
104,487,111
21.702
7.780
i ^72
1482.83
17497455
222,784
105,154,048
21.725
7.786
473
1485.97
175716.75
223,729
105,823,817
21.749
7.791
474
1 489. 1 1
176460.45
224,676
166,496,424
21.771
7.797
1^
MATHEMATICAL TABLES.
Number,
or
Circum
Circular
Soimrc
Cube.
Square
Cube
Diameter.
ference.
Area.
fcj*l >M1 i ^*
Root.
Root.
475
1492.26
177205.87
225,625
107,171,875
21.794
7.802
476
1495.36
177952.79
226,576
107,850,176
21.817
7.808
477
1498.54
178701.27
227,529
108,531,333
21.840
7.813
478
1501.68
179451.33
228,484
109,215,352
21.863
7.819
479
1504.82
180202.96
229,441
109,902,239
21.886
7.824
480
1507.96
180956.16
230,400
110,592,000
21.909
7.830
481
I5II.IO
181712.92
231,361
111,284,641
21.932
7.835
482
1514.25
182467.26
232,324
111,980,168
21.954
7.840
483
1517.39
183225.18
233,289
112,678,587
21.977
7.846
484
1520.53
183984.66
234,256
"3,379,904
22.000
7.851
485
1523.67
184745.71
235,225
114,084,125
22.023
7.857
486
1526.81
185508.33
236,196
114,791,256
22.045
7.862
487
1529.95
186272.53
237,169
115,501,303
22.069
7.868
488
1533.90
187038.29
238,144
116,214,272
22.091
7.873
489
1536.24
187805.63
239,121
116,936,169
22.113
7.878
490
1539.38
188574,54
240,100
117,649,000
22.136
7.884
491
1542.52
189345.01
241,081
118,370,771
22.158
7.889
492
1545.66
I9OII7.06
242,064
119,095,488
22.181
7.894
493
1548.80
190890.68
243,049
119,823,157
22.204
7.899
494
1551.95
191665.87
244,036
120,553,784
22.226
7.905
495
155509
192442.63
245,025
121,287,375
22.248
7.910
496
1558.23
193220.96
246,016
122,023,936
22.271
7.915
497
1561.37
194000.86
247,009
122,763,473
22.293
7.921
498
1564.51
194782.34
248,004
123,505,992
22.316
7.926
499
1567.55
195565.38
249,001
124,251,499
22.338
7.932
500
1570.80
196350.00
250,000
125,000,000
22.361
7.937
501
1573.94
I97136.18
251,001
125,751,501
22.383
7.942
502
1577.08
197923.94
252,004
126,506,008
22.405
7.947
503
1580.22
198713.26
253,009
127,263,527
22.428
7.953
504
1583.36
199504.16
254,016
128,024,864
22.449
7.958
505
1586.50
200296.63
255,025
128,787,625
22.472
7963
506
1589.64
201090.67
256,036
129,554,216
22.494
7.969
507
1592.79
201886.28
257,049
130,323,843
22.517
7.974
508
1595.93
202683.46
258,064
131,096,512
22.539
7.979
509
1599.07
203487.70
259,081
131,872,229
22.561
7.984
510
1602.21
204282.54
260,100
132,651,000
22.583
7.989
5"
1605.35
205084.43
261,121
133,432,831
22.605
7.995
512
1608.49
205887.84
262,144
134,217,728
22.627
8.000
513
161I.64
206692.93
263,169
135,005,697
22.649
8.005
514
1614.78
207499.53
264,196
135,796,744
22.671
8.010
515
1617.92
208307.71
265,225
136,590,875
22.694
8.016
516
1621.06
209117.46
266,256
137,388,096
22.716
8.021
517
1624.20
209928.78
267,289
138,188,413
22.738
8.026
S18
1627.34
210741.66
268,324
138,991,832
22.759
8.031
519
1630.49
2II556.I2
269,361
139,798,359
22.782
8.036
520
1633.63
212372.16
270,400
140,608,000
22.803
8.041
521
1636.77
213189.76
271,441
141,420,761
22.825
8.047
522
1639.93
214008.93
272,484
142,236,648
22.847
8.052
NUMBERS, OR DIAMETERS OF CIRCLES, &c.
77
' Nnmber,
or
OrcoiB
Circular
Square.
Cube,
Square
Cube
iDofDClCT.
Area.
^0^mm^^0^
Root.
Root.
S»3
1643.05
214829.67
273,529
143,055,667
22.869
8.057
5«4
1646.19
215651.99
274,576
143,877,824
22.891
8.062
5»5
1649.34
21647587
275,625
144,703,125
22.913
8.067
526
1652.48
217301.33
276,676
145,531,576
22.935
8.072
527
1655.62
218128.35
277,729
146,363,183
22.956
8.077
5*8
1658.76
218956.95
278,784
147,197,952
22.978
8.082
5*9
1661.90
219787.12
279,841
148,035,889
23.000
8.087
530
1665.04
220618.86
280,900
148,877,000
23.022
8.093
531
1668.18
221452.16
281,961
149,721,291
23.043
8.098
' 53*
167133
222287.04
283,024
150,568,768
23.065
8.103
533
1674,47
223123.50
284,089
151,419,437
23087
8.108
534
* 1677.61
223961.52
285,156
152,273,304
23.108
8.II3
535
1680.75
224801. II
286,225
153,130,375
23.130
8. 118
536
1683.80
225642.27
287,296
153,990,656
23.152
8.123
537
I 1687.04
226487.01
288,369
154,854,153
23173
8.128
538
. 1690.18
2273293^
289,444
155,720,872
23195
8.133
539
1 1693.32
228175.19
290,521
156,590,819
23.216
8.138
540
1696.46
229022.64
291,600
157,464,000
23238
8.143
541
1699.60
229871.65
292,681
158,340,421
23259
8.148
542
1702.74
230722.24
293,764
159,220,088
23.281
8.153
543
1705.88
231574.40
294,849
160,103,007
23.302
8.158
544
1709.03
232428.13
295,936
160,989,184
23324
8.163
545
1712.17
233283.43
297,025
161,878,625
23345
8.168
546
' 171531
234140.30
298,116
162,771,336
23367
8.173
' 547
1718.45
234998.74
299,209
163,667,323
23388
8.178
548
. 1721.59
235858.76
300,304
164,566,592
23409
8.183
549
• 1724.73
236720.34
301,401
165,469,149
23431
8.188
550
' 1727.88
23758350
302,500
166,375,000
23452
8.193
551
1731.02
238448.22
303,601
167,284,151
23473
8.198
55*
1734.16
239314.52
304,704
168,196,608
23495
8.203
' 553
1737.30
240182.38
305,809
169,112,377
235^6
8.208
554
I 1740.44
241051.82
306,916
170,031,464
23537
8.213
555
1743.58
241922.83
308,025
170,953,875
23558
8.218
556
1746.72
242795.41
309,136
171,879,616
23579
8.223
557
1749.77
243669.56
310,249
172,808,693
23.601
8.228
558
1753.09
244545.28
311,364
173,741,112
23.622
8.233
559
1756.^^5
245422.57
312,481
174,676,879
23643
8.238
560
1759.29
246301.44
313,600
175,616,000
23.664
8.242
561
1762.43
247181.87
314,721
176,558,481
23.685
8.247
562
! 1765.57
248063.87
315,844
177,504,328
23.706
8.252
563
1768.72
248947.45
316,969
178,453,547
23.728
8.257
564
1771.86
249832.59
318,096
179,406,144
23.749
8.262
565
1775.00
25071931
319,225
180,362,125
23.769
8.267
566
1778.14
251607.60
320,356
181,321,496
23.791
8.272
567
1781.28
252497.36
321,489
182,284,263
23.812
8.277
568
1 1784.42
253388.88
322,624
183,250,432
23833
8.282
569
1787.57
254281.88
323,761
184,220,009
23854
8.286
570
1
1790.71
255176.64
324,900
185,193,000
23875
8.291
78
MATHEMATICAL TABLES.
Number,
or
Circum
Circular
Square.
Cube.
Square
Cube
Diameter.
ference.
Area.
Root.
Root.
571
179385
256072.60
326,041
186,169,411
23.896
8.296
572
1796.99
256970.31
327,184
187,149,248
23.916
8.301
573
1800.13
257869.59
328,329
188,132,517
23937
8.306
574
1803.27
258770.45
329,476
189,119,224
23.958
8.3 II
575
1806.42
259672.87
330,625
190,109,375
23979
8.315
576
1809.56
260576.87
331,776
191,102,976
24.000
8.320
577
1812.80
261482.43
332,929
192,100,033
24.021
8.325
578
1815.84
262388.57
334,084
193,100,552
24.042
8.330
579
1818.98
263298.28
335,241
194,104,539
24.062
8.335
580
1822.12
264208.56
336,400
195,112,000
24,083
8.339
581
1825.26
265120.46
337,561
196,122,941
24.104
8.344
582
1828.41
266033.82
338,724
197,137,368
24.125
8.349
583
1831.55
266948.82
339,889
198,155,287
24.145
8.354
584
1834.69
267865.38
341,056
199,176,704
24.166
8.359
585
1837.83
268783.57
342,225
200,201,625
24,187
8.363
586
1840.97
269703.21
343,396
20i;23o,o56
24.207
8.368
587
1 844. 1 1
270624.49
344,569
202,262,003
24,228
8.373
588
1847.26
271547.33
345,744
203,297,472
24.249
8.378
589
1850.40
272471.75
346,921
204,336,469
24.269
8.382
590
185354
273397.74
348,100
205,379,000
24.289
8.387
591
1856.68
274325.29
349,281
206,425,071
1 24.310
8.392
592
1859.82
275254,42
350,464
207,474,688
24.331
8.397
593
1862.96
276185.12
351,649
208,527,857
24.351
8.401
594
1 866. 11
277117.39
352,836
209,584,584
24.372
8.406
595
1869.25
278051.23
354,025
210,644,875
24393
8.4II
596
1872.39
278986.64
355,216
211,708,736
24.413
8.415
597
1875.53
279923.62
356,409
212,776,173
24.433
8.420
598
1878.67
280862.18
357,604
213,847,192
24.454
8.425
599
1881.81
281802.30
358,801
214,921,799
24.474
8.429
600
1884.96
282744.00
360,000
216,000,000
24.495
8.434
601
1888.10
283687.26
361,201
217,081,801
24.515
8.439
602
1891.24
284632.10
362,404
218,167,208
24.536
8.444
603
1894.38
285578.50
363,609
219,256,227
24.556
8.448
604
1897.52
286526.48
364,816
220,348,864
24.576
8.453
605
1900.66
287476.03
366,025
221,445,125
24.597
8.458
606
1903.80
288426.15
367,236
222,545,016
i 24.617
8.462
607
1906.95
289379.84
368,449
223,648,543
24.637
8.467
608
1910.09
290334.10
369,664
224,755,712
24.658
8.472
609
1913.23
291289.93
370,881
225,866,529
24.678
8.476
610
1916.37
2^)2247.34
372,100
226,981,000
24.698
8.481
611
1919.51
293206.31
373,321
228,099,131
24.718
8.485
612
1922.65
294166.85
374,544
229,220,928
24.739
8.490
613
1925.80
295128.97
375,769
230,346,397
24.758
8.495
614
1928.94
296092.65
376,996
231,475,544
24.779
8.499
615
1932.08
297057.91
378,225
232,608,375
24.799
8.504
616
193522
298024.74
379,456
233,744.896
' 24.819
8.509
617
1938.36
298993.14
380,689
234,885,113
24839
8.513
618
1941.50
299963.00
381,924
236,029,032
1 24.859
8.518
NUMBERS, OR DIAMETERS OF CIRCLES, Sec
79
Nomber,
Dbm«ter.
CSrcuin
ferenc«.
Circular
Area.
Square.
Cube.
Square
Root.
Cube
Root.
6io
1 ^
1944.65
300934,64
383,161
237,176,659
24.879
8.522
620
1947.79
301907.76
384,400
238,628,000
24.899
8.527
621
1
1950.93
302S82.44
3^5Mx
239,483,061
24.919
8532
622
195407
303858.69
386,884
240,641,848
24.939
8.536
6^3
1957.21
304836.51
388,129
241,804,367
24.959
8.541
624
1960.35
305815.91
389,376
242,970,624
24.980
8.545
625
1963.50
306796.87
390,625
244,140,625
25.000
8.549
626
1966.64
307779.41
391,876
245,314,376
25.019
8.554
627
1969.78
308763.41
393,129
246,491,883
25.040
8.559
628
1972.92
309749.19
394,384
247,673,152
25059
8.563
629
1976.06
310736.44
395,641
248,858,189
25.079
8.568
630
1979.20
311725.26
396,900
250,047,000
25.099
8.573
631
1982.34
312715.64
398,161
25^239,591
25.119
8.577
' 632
1985.49
313707.58
399.424
252,435,968
25.139
8.582
^3S
1988.63
31470I.I4
400,689
253,636,137
25.159
8.586
634
1991.77
315696.64
401,956
254,840,104
25179
8.591
635
1994,91
316692.91
403,225
256,047,875
25.199
8.595
636
1998.05
31769I.15
404,496
257,259,456
25.219
8599
637
2001.19
318690.97
405,769
258,474,853
25.239
8.604
638
2004.34
319692.35
407,044
259,694,072
25.259
8.609
639
2007.48
320695.31
408,321
260,917,119
25.278
8.613
640
2010.62
321699.84
409,600
262,144,000
25.298
8.6i8
641
2013.76
322705.93
410,881
263,374,721
25.318
8.622
642
2016.90
323713.60
412,164
264,609,288
25.338
8.627
643
2020.04
324722.84
413,449
265,847,707
25357
8.631
644
2023.19
32573365
414,736
267,089,984
25377
8.636
645
2026.33
326746.03
416,025
268,836,125
25397
8.640
646
2029.47
327759.98
417,316
269,586,136
25.416
8.644
647
2032.61
328775.50
418,609
270,840,023
25.436
8.649
648
2035.76
329792.60
419,904
272,097,792
25.456
8.653
649
2038.89
330811.26
421,201
273,359,449
25.475
8.658
650
2042.04
331831.50
422,500
274,625,000
25495
8.662
1 651
2045.18
332853.40
423,801
275,894,451
25515
8.667
652
2048.32
333876.68
425,104
277,167,808
25534
8.671
653
2051.46
334901.62
426,409
278,445,077
25554
8.676
654
2054.60
335928.14
427,716
279,726,264
25573
8.680
' 655
2057.74
336956.23
429,025
281,011,375
25.593
8.684
656
2060.88
3379S5.89
430,336
282,800,416
25.612
8.689
657
2064.03
339017.12
431,649
283,593,393
25.632
8.693
658
2067.17
340049.92
432,964
284,890,312
25651
8.698
659
2070.31
341084.29
434,281
286,191,179
25.671
8.702
660
2073.45
342120.24
435,600
287,496,000
25690
8.706
661 .
2076.59
34315775
436,921
288,804,781
1 25.710
8.711
662
2079.73
344196.33
438,244
290,117,528
25.720
8.715
663
2082.88
345237.49
439,569
291,434,247
1 25.749
8.719
664
2086.02
346279.71
440,896
292,754,944
25.768
8.724
665
2089.16
34732351
442,225
294,079,625
25787
8.728
666
2092.30
^348368.88
443,556
295,408,296
25.807
8.733
8o
MATHEMATICAL TABLES.
Number,
or
Circum
Circular
Sauare.
Cube.
Square
Cube
Diameter.
ference.
Area.
M^^i^KA^hM ^r»
Root.
Root.
667
2095.44
349416.40
444,889
296,740,963
25.826
8.737
668
2098.58
350464.32
446,224
298,077,632
25.846
8.742
669
2101.73
35151430
447,561
299,418,309
25.865
8.746
670
2104.87
352566.06
448,900
300,763,000
25.884
8.750
671
2108.01
353619.28
450,241
302,111,711
25.904
8.753
672
2III.15
354674.07
451,584
303,464,448
25923
8.759
673
2114.29
35573043
452,929
304,821,217
25.942
8.763
674
211743
35678837
454,276
306,182,024
25.961
8.768
675
2i2a58
357847.87
455,625
307,546,875
25.981
8.772
676
2123.72
358908.95
456,976
308,915,776
26.000
8.776
677
2126.86
359971.59
458,329
310,288,733
26.019
8.781
678
2130.00
361035.81
459,684
311,665,752
26.038
8.785
679
213314
362101.60
461,041
313,046,839
26.058
8.789
680
2136.28
363168.96
462,400
314,432,000
26.077
8.794
681
2139.42
364237.88
463,761
315,821,241
26.096
8.798
682
2142.57
365308.38
465,124
317,214,568
26.115
8.802
683
2145.71
366380.40
466,489
318,611,987
26.134
8.807
684
2148.85
367454.10
467,856
320,013,504
26.153
8.81 1
685
2151.99
368529.31
469,225
321,419,125
26.172
8.815
686
215513
369600.60
470,596
322,828,856
26.192
8.819
687
2158.27
370684.45
471,969
324,242,703
26.211
8.824
688
2161.42
371764.37
473*344
325,660,672
26.229
8.828
689
2164.56
372845.87
474,721
327,082,769
26.249
8.832
690
2167.70
373928.94
476,100
328,509,000
26.268
8.836
1 691
2170.84
37501357
477,481
329,939*371
26.287
8.841
692
2173.98
376099.78
478,864
331,373,888
26.306
8.845
693
2177.12
37718756
480,249
332,812,557
26.325
8.849
694
2180.27
378276.91
481,636
334,255,384
26.344
8.853
695
2183.41
37936783
483,025
335,702,375
26.363
8.858
696
2186.55
380460.32
484,416
337,153,536
26.382
8.862
697
2189.69
38155438
485,809
338,608,873
26.401
8.866
698
2192.83
382650.02
487,204
340,068,392
26.419
8.870
699
2195.97
383747.22
488,601
341,532,099
26.439
8.875
700
2199.12
384846.00
490,000
343,000,000
26.457
8.879
701
2202.26
38594952
491,401
344,472,101
26.476
8.883
702
2205.40
387048.26
492,804
345,948,088
26.495
8.887
703
2208.54
388151.74
494,209
347,428,927
26.514
8.892
704
2211.68
389256.80
495,616
348,913,664
26.533
8.896
705
2214.82
39036343
497,025
350,402,625
26.552
8.900
yo6
2217.96
39147163
498,436
351,895,816
26.571
8.904
707
2221. II
392581.40
499,849
353,393,243
26.589
8.908
708
2224.25
393692.74
501,264
354,894,912
26.608
8.913
709
2227.39
394805.65
502,681
356,400,829
26.627
8.917
710
2230.53
395920.14
504,100
357,911,000
26.644
8.921
711
2233,67
397036.19
505,521
359,425,431
26.664
8.925
712
2236.81
398151.81
506,944
360,944,128
26.683
8.929
713
2239.96
399273.01
508,369
362,467,097
26.702
8.934
714
2243.10
400393.73
509,796
363,994,344
26.721
8.938
NUMBERS, OR DIAMETERS OF CIRCLES, &c.
8l
I Kiunber,
or
Diameter.
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
Grcum
ference.
2246.24
2249.38
2252.52
2255.66
2258.81
2261.95
2265.09
226S.23
2271.37
227451
2277.66
2280.80
2283.94
2287.08
2290.22
2293.36
2296.50
2299.65
2302.79
2305.93
230907
2312.21
2315.35
2318.50
2321.64
2324.78
2327.92
2331.06
2334.20
2337.35
2340.49
2343.63
2346.77
2349.91
2353.05
2356.20
2359.34
2362.48
2365.62
2368.76
2371.90
2375.04
2378.19
2381.33
2384.47
2387.61
2390.75
239389
Grcular
Area.
4OI516.II
402640,02
403765.50
404892.54
406021.16
407151.36
408283.32
409416.45
410551.25
411687.93
412825.87
413965.24
415106.06
416249.43
41739376
418539.66
419687.12
420836.14
421986.78
423138.96
424292.71
425442.03
426604.93
42776339
428923.43
430085.04
431248.21
432412.96
43357928
434747.17
435916.63
437087.66
438260.26
43943448
440610.18
441787.50
442966.38
444146.84
445328.86
446512.46
44769763
448884.37
450072.68
451262.56
452454.01
453647.04
454841.63
456037.87
Square.
5ii>225
512,656
514,089
515*524
516,961
518,400
519,841
521,284
522,729
524>i76
525*625
527,076
528,529
529*984
531,441
532,900
534,361
535,824
537,289
538,756
540,225
541,696
543,169
544,644
546,121
547,600
549,081
550,564
552,049
553,536
555,025
556,516
558,009
559,504
561,001
562,500
564,001
565,504
567,009
568,516
570,025
571,536
573,049
574,564
576,081
577,600
579,^21
580,644
Cube.
365,525,875
367,061,696
368,601,813
370,146,232
371,694,959
373,248,000
374,805,361
376,367,048
377,933,067
379,503,424
381,078,125
382,657,176
384,240,583
385,828,352
387,420,489
389,017,000
390,617,891
392,223,168
393,832,837
395,446,904
397,065,375
398,688,256
400,315,553
401,947,272
403,583,419
405,224,000
406,869,021
408,518,488
410,172,407
411,830,784
413,493,625
415,160,936
416,832,723
418,508,992
420,189,749
421,875,000
423,564,751
424,525,900
426,957,777
428,661,064
430,368,875
432,081,216
433,798,093
435,519,512
437,245,479
438,976,000
440,711,081
442,450,728
Square
Cube
Root.
Root.
26.739
8.942
26.758
8.946
26.777
8.950
26.795
8.954
26.814
8.959
26.833
8.963
26.851
8.967
26.870
8.971
26.889
8.975
26.907
8.979
26.926
^.983
26.944
8.988
26.963
8.992
26.991
8.996
27.000
9.000
27.018
9.004
27.037
9.008
27.055
9.012
27.074
9.016
27.092
9.020
27. Ill
9023
27.129
9.029
27.148
9033
27.166
9037
27.184
9.041
27.203
9045
27.221
9.049
27.239
9053
27.258
9057
27.276
9.061
27.295
9.065
27.313
9.069
27.331
9073
27.349
9077
27.368
9.081
27.386
9.086
27.404
9.089
27.423
9.094
27.441
9.098
27.459
.9.102
27.477
9.106
27.495
9.109
27514
9.II4
27532
9. 1 18
27.549
9.122
27.568
9.126
27.586
9.129
27.604
9134
iV
82
MATHEMATICAL TABLES.
Number,
or
Diameter.
Circum
ference.
Circular
Area.
Square.
Cube.
Square
Root.
Cube
Root,
763
2397.04
457235.53
582,169
444,194,947
27.622
9.138
764
2400.18
458435.83
583,696
445,943,744
27.640
9.142
765
2403.32
459635.71
585*225
447,697,125
27.659
9.146
766
2406.46
460838.16
586,756
449,455,096
27.677
9.149
767
2409.60
462042.18
588,289
451,217,663
27.695
9.154
768
2412.74
463247.76
589,824
452,984,832
27.713
9.158
769
2415.98
464454.92
59i»36i
454,756,609
27.731
9.162
770
2419.03
465663.66
592,900
456,533,000
27.749
9.166
771
2422.17
466873.96
594,441
458,314,011
27.767
9.169
772
242531
468085.83
595,984
460,099,648
27.785
9.173
77^
2428.45
469299.27
597,529
461,889,917
27.803
9.177
774
243159
470514.29
599,076
463,684,824
27.821
9.181
775
243474
471730.87
600,625
465,484,375
27.839
9.185
776
2437.88
47^2949.03
602,176
467,288,576
27.857
9.189
777
2441.02
474168.75
603,729
469,097,433
27.875
9.193
778
2444.16
475396.05
605,284
470,910,952
27.893
9.197
779
2447.30
476612.92
606,841
472,729,139
27.910
9.201
780
2450.44
477837.36
608,400
474,552,000
27.928
9.205
781
2453.58
479063.36
609,961
476,379,541
27.946
9.209
782
2456.73
480290.94
611,524
478,211,768
27.964
9.213
783
2459.87
481520.10
613,089
480,048,687
27.982
9.217
784
2463.01
482750.82
614,656
481,890,304
28.000
9.221
785
2466.15
483983.11
616,225
483,736,025
28.017
9.225
786
2469.29
485216.97
617,796
485,587,656
28.036
9.229
787
2472.43
486452.41
619,369
487,443,403
28.053
9.233
788
2475.48
487689.73
620,944
489,303,872
28.071
9.237
789
2478.72
488927.99
622,521
491,169,069
28.089
9.240
790
2481.86
490168.14
624,100
493,039,000
28.107
9.244
791
2485.00
491409.85
625,681
494,913,671
28.125
9.248
792
2488.14
492653.14
627,264
496,793,088
28.142
9.252
793
2491.28
493898.20
628,849
498,677,257
28.160
9.256
.794
2494.43
495144.43
630,436
500,566,184
28.178
9.260
795
2497.57
496392.43
632,025
502,459,875
28.196
9.264
796
2500.71
497648.40
633,616
504,358,336
28.213
9.268
797
2503.85
498893.14
635,209
506,261,573
28.231
9.271
798
2506.99
500145.86
636,804
508,169,592
28.249
9.275
799
2510.13
501400.14
638,401
510,082,399
28.266
9.279
800
2513.28
502656.00
640,000
512,000,000
28.284
9.283
801
2516.42
503913.42
641,601
513,922,401
28.302
9.287
802
2519.56
505172.43
643,204
515,849,608
28.319
9.291
803
2522.70
506432.98
644,809
517,781,627
28.337
9.295
804
2525.84
507655.52
646,416
519,718,464
28.355
9.299
805
2528.98
508958.83
648,025
521,660,125
28.372
9.302
806
2532.12
510224.II
649,636
523,606,616
28.390
9.306
807
2535.27
511490.96
651,249
525,557,943
28.408
9.310
808
2538.41
512759.38
652,864
527,514,112
28.425
9.314
809
2541.55
514029.37
654,481
529,474,129
28.443
9.318
810
2544.09
515300.94
656,100
531,441,000
28.460
9.321
NUMBERS, OR DIAMETERS OF CIRCLES, &c.
83
Number,
or
Crcum
Circular
Square.
Cube.
Square
Cube
Dianwtw.
ference.
Area.
Root.
Root.
i 811
2547.83
516574.07
657,721
533,411,731
28.478
9325
812
255097
517848.77
659,344
535,387,328
28.496
9329
813
255412
519125.05
660,969
537,366,797
28.513
9333
814
2557.26
520402.85
662,596
539,353,144
28.531
9337
815
2560.40
521682.31
664,225
541,343,375
28.548
9341
S16
2563.54
522663.30
665,856
543,338,496
28.566
9345
817
2566.68
524245.86
667,489
545,338,513
28.583
9348
818
2569.82
525529.98
669,124
547,343,432
28.601
9.352
819
2572.97
526815,68
670,761
549,353,259
28.618
9356
820
2576.11
528102.96
672,400
551,368,000
28.636
9.360
821
2579.25
529391.80
674,041
553,387,6^1
28.653
9364
822
2582.39
530682.21
675,684
555,412,248
28.670
9367
823
2585.53
531974.39
677,329
557,441,767
28.688
9.371
824
2588.64
533267.75
678,976
559,476,224
28.705
9375
825
2591.82
534562.87
680,625
561,515,625
28.723
9.379
826
1
2594.96
535859.57
682,276
563,559,976
28.740
9383
8a7
2598.10
53715983
683,929
565,609,283
28.758
9386
828
2601.24
53845762
685,584
567,663,552
28.775
9.390
829
260438
539759.08
687,241
569,722,789
28.792
9394
830
2607.52
541062.06
688,900
571,787,000
28.810
9.398
831
2610.66
542366.60
690,561
573.856,191
28.827
9.401
83*
2613.81
543672.72
692,224
575,930,368
28.844
9.405
833
2616.95
544980.52
693,889
578,009,537
28.862
9.409
834
2620.09
546289.68
695,556
580,093,704
28.879
9.413
i 835
2623.23
547600.51
697,225
582,182,875
28.896
9.417
836
2626.37
548912.91
698,896
584,277,056
28.914
9.420
837
2629.51
550226.89
700,569
586,376,253
28.931
9.424
838
2632.64
551542.43
702,244
588,480,472
28.948
9.428
839
263S.80
55285958
703,921
590,589,719
28.965
9432
840
2638.94
554178.24
705,600
592,704,000
28.983
9435
84,
2642.08
555498.49
707,281
594,823,321
29.000
9439
, 842
2645.22
556820.32
708,964
596,947,688
29.017
9443
! 843
2648.36
558143.72
710,649
599,077,107
29.034
9447
844
2651.51
559468.69
712,336
601,211,584
29.052
9450
84s
2654.65
560795.23
714,025
603,351,125
29.069
9454
846
2657.79
562123.34
715,716
605,495,736
29.086
9.458
, 847
2660.93
563456.82
717,409
607,645,423
29.103
9.461
1 848
2664,07
564784.28
719,104
609,800,192
29.120
9.465
849
2667.21
566117.IO
720,801
611,960,049
29.138
9469
, 850
2670.36
56745159
722,500
614,125,000
29.155
9473
1 851
2673.50
568787.46
724,201
616,295,051
29.172
9476
852
' 2676.64
570125.00
725,904
618,470,208
29.189
9.480
' 853
2679.78
571464.10
727,609
620,650,477
29.206
9483
' 854
2682.92
572804.78
729,316
622,835,864
29.223
9.487
85s
2686.06
574147.03
731,025
625,026,375
29.240
9.491
' 856
2689.20
575490.85
732,736
627,222,016
29257
9495
' ^57
2692.35
576836.24
734,449
629,422,793
29274
9499
i 858
2695.49
578183.20
736,164
631,628,712
29.292
9.502
84
MATHEMATICAL TABLES.
Number,
or
Circum
Circular
Square.
Cube.
Square
Cube
Diameter.
ference.
Area.
Root.
Root.
859
2698.63
579531.73
737,881
633,839,779
29.309
9.506
860
2701.77
580881.84
739,600
636,056,000
29.326
9509
861
2704.91
582233.51
741,321
638,277,381
29.343
9.513
862
2708.05
583586.75
743,044
640,503,928
29.360
9517
863
2711.20
584941.57
744,769
642,735,647
29.377
9.520
864
271434
586297.95
746,496
644,972,544
29394
9524
865
2717.48
58765591
748,225
647,214,625
29.411
9.528
866
2720.66
589015.41
749,956
649,461,896
29.428
9532
867
2723.76
590376.54
751,689
651,714,363
29445
9535
868
2726.90
591739.20
753,424
653,972,032
29.462
9.539
869
2730.05
5<>3 10344
755,161
656,234,909
29.479
9543
870
273319
594469.26
756,900
658,503,000
29.496
9546
871
2736.33
595836.44
758,641
660,776,311
29.513
9550
872
2739.87
597205.59
760,384
663,054,848
29.529
9554
873
2742.61
598576.91
762,129
665,338,617
29.546
9557
874
274575
599948.21
763,876
667,627,624
29.563
9.561
875
2748.90
601321.87
765,625
669,921,875
29.580
9565
876
2752.04
602697,11
767,376
672,221,376
29597
9.568
877
275518
604073.91
769,129
674,526,133
29.614
9572
878
2758.32
605451.49
770,884
676,836,152
29.631
9.575
879
2761.46
606832.24
772,641
679,151,439
29.648
9.579
880
2764.60
608213.76
774,400
681,472,000
29.665
9.583
881
2767.74
609596.84
776,161
683,797,841
29.682
9.586
882
2770.89
610981.50
777,924
686,128,968
29.698
9590
883
2774.03
612367.74
779,689
688,465,387
29715
9.594
884
2777.17
61375554
781,456
690,807,104
29.732
9597
885
2780.31
615144.91
783,225
693,154,125
29749
9.601
886
278345
61653585
784,996
695,506,456
29.766
9.604
887
2786.59
617928.37
786,769
697,864,103
29.782
9.608
888
278975
619322.45
788,544
700,227,072
29.799
9.612
889
2792.88
620718.11
790,321
702,595,369
29.816
9615
890
2796.02
622115,34
792,100
704,969,000
29.833
9.619
891
2799.16
623514.13
793,881
707,347,971
29.850
9.623
892
2802.30
624914.50
795,664
709,732,288
29.866
9.626
893
2805.44
626316.44
797,449
712,121,957
29.883
9.630
894
2808.59
62771995
799,236
714,516,984
29.900
9.633
895
2811.73
629120.35
801,025
716,917,375
29.916
9637
896
2814.87
630531.68
802,816
719,323,136
29933
9.640
897
2818.82
63193990
804,609
721,734,273
29.950
9.644
898
2821.15
63334970
806,404
724,150,792
29.967
9.648
899
2824.29
634768.13
808,201
726,572,699
29.983
9651
900
2827.44
636174.00
810,000
729,000,000
30.000
9.655
901
2830.58
637588.50
811,804
731,432,701
30.017
9.658
902
2833.72
639004.58
813,604
733,870,808
30.033
9.662
903
2836.86
640422.22
815,409
736,314,327
30.050
9.666
904
2840.00
641841.44
817,216
738,763,264
30.066
9.669
905
2843.14
643262,23
819,025
741,217,625
30.083
9.673
906
2846.28
644684.74
820,836
743,677,416
30.100
9.676
NUMBERS, OR DIAMETERS OF CIRCLES, &c.
85
Xumber,
or
Grcum
ference.
907
2849.43
908
2852.57
909
285571
910
2858.85
911
2861.99
912
2865.13
913
2868.29
914
2871.42
915
2874.56
916
2877.70
917
2880.84
918
2883.98
919
2887.13
920
2890.27
921
2893.41
922
2896.55
923
2899.69
924
2902.83
925
2905.98
926
2909.12
927
2912.26
928
2915.40
929
2918.54
930
2921.68
931
2924.82
932
2927.97
933
2931. II
934
2934.25
935
2937.39
936
2940.53
937
2943.67
938
2946.82
939
2949.96
940
2953.10
941
2956.24
942
2959.38
943
2962.43
944
2965.67
945
2968.81
946
2971.95
947
2975.09
948
2978.23
949
2981.37
950
2984.52
951
2987.66
952
2990.72
953
299394
954
2997.08
Ctrcttlar
Area.
Square.
646108.52
822,649
64753402
824,464
648961.09
826,281
650389.74
828,100
651819.95
829,921
65325173
831,744
654689.09
833,569
656120.81
835,396
657556.51
837,225
658994.58
839,056
660432.22
840,889
661875.42
842,724
663318.20
844,561
664762.56
846,400
666208.48
848,241
66765597
850,084
669IOI.61
851,929
67055567
853,776
672007.87
855,625
673461.65
857,476
674916.99
859,329
67637391
861,184
677832.40
863,041
679292.46
864,900
680754.08
866,761
682217.30
868,624
683682.06
870,489
685148.40
872,356
686616.31
874,225
688085.79
876,096
689556.85
877,969
691029.47
879,844
692503.67
881,721
69397944
883,600
695456.77
885,481
69693568
887,364
698416.14
889,249
699898.21
891,136
701381.83
893,025
702867.02
894,916
704350.25
896,809
705841.80
898,704
707332.02
900,601
708023.50
902,500
710316.54
904,401
7I181I.16
906,304
71330734
908,209
714805.10
910,116
Cube.
746,142,643
748,613,312
751,089,429
753,571,000
756,058,031
758,550,528
761,048,497
763,551,944
766,060,875
768,575,296
771,095,213
773,620,632
776,151,559
778,688,000
781,229,961
783,777,448
786,330,467
788,889,024
791,453,125
794,022,776
796,597,983
799,178,752
801,765,089
804,357,000
806,954,491
809,557,568
812,166,237
814,780,504
817,400,375
820,025,856
822,656,953
825,293,672
827,936,019
830,584,000
833,237,621
835,896,888
838,561,807
841,232,384
843,908,625
846,590,536
849,278,123
851,971,392
854,670,349
857,375,000
860,085,351
862,801,408
865,523,177
868,250,664
Square
Cube
Root.
Root.
30.116
9.680
30.133
9.683
30.150
9.687
30.163
9.690
30.183
9694
30.199
9.698
30.216
9.701
30.232
9705
30.249
9.708
30.265
9.712
30.282
9.715
30.298
9.718
30.315
9.722
30.331
9.726
30.348
9.729
30.364
9.733
30.381
9736
30.397
9.740
30.414
9743
30.430
9747 .
30.447
9750
30.463
9754
30.479
9757
30.496
9.761
30.512
9.764
30.529
9.768
30.545
9.771
30.561
9.775
30.578
9.778
30.594
9.783
30.610
9.785
30.627
9.789
30.643
9.792
30.659
9.796
30.676
9.799
30.692
9.803
30.708
9.806
30.724
9.810
30.741
9.813
30.757
9.817
30.773
9.820
30.790
9.823
30.806
9.827
30.822
9.830
30.838
9.834
30.854
9.837
30.871
9.841
30.887
9.844
86
MATHEMATICAL TABLES.
j Number,
or
Circum
Circular
Sauaie.
Cube.
Square
Cube
Diameter.
ference.
Area.
»i^*i **•*• *p*
Root.
Root.
955
3000.22
716304.43
912,025
870,983,875
30.903
9.848
956
300336
717805.33
913,936
873,722,816
30.919
9.851
957
3006.51
719307.80
915,849
876,467,493
30.935
9.854
958
3009.65
720811.84
917,764
879,217,912
30.951
9.858
959
3012.79
722317.45
919,681
881,974,079
30.968
9.861
960
301593
723824.64
921,600
884,736,000
30.984
9.865
961
3019.07
72533339
923»52i
887,503,681
31.000
9.868
962
3022.21
726843.71
925,444
890,277,128
31.016
9.872
963
3025.36
72835561
927,369
893,056,347
31.032
9875
964
3028.50
729869.07
929,296
895,841,344
31.048
9.878
965
3031.64
73 1384. 1 1
931,225
898,632,125
31.064
9.881
966
3034.78
732900.72
933,156
901,428,696
31.080
9.885
967
3037.92
734418.90
935,089
904,231,063
31.097
9.889
968
3041.06
735938.64
937,024
907,039,232
3^'^^3
9.892
969
3044.21
737459.96
938,961
909,853,209
31.129
9.895
970
3047.35
738982.86
940,900
912,673,000
31.145
9.899
971
3050.49
740507.32
942,841
915,498,611
31.161
9.902
972
3053.63
742033.35
944,784
918,330,048
31.177
9.906
973
3056.77
743560.95
946,729
921,167,317
31.193
9909
• 974
3059.91
745090.13
948,676
924,010,424
31.209
9.912
975
3063.06
746620.87
950,625
926,859,375
31.225
9.916
976
3066.20
748153.19
952,576
929,714,176
31.241
9.919
977
3069.36
749687.07
954,529
932,574,833
31257
9923
978
3072.48
751222.53
956,484
935,441,352
31.273
9.926
979
3075.62
75275956
958,441
938,313,739
31.289
9.929
980
3078.76
754298.16
960,400
941,192,000
31305
9933
981
3081.90
755838.32
962,361
944,076,141
31321
9936
982
3085.05
757380.06
964,324
946,966,168
31.337
9.940
983
3088.19
758923.38
966,289
949,862,087
31.353
9943
984
3091.33
760468.26
968,256
952,763,904
31.369
9.946
985
3094.47
762014.71
970,225
955,671,625
31385
9.950
986
309761
763562.73
972,196
958,585,256
31.401
9953
987
3100.75
76511933
974,169
961,504,803
31.416
9956
988
3103.96
766663.49
976,144
964,430,272
31.432
9.960
989
3107.04
768216.23
978,121
967,361,669
31.448
9963
990
3IIO.18
769770.54
980,100
970,299,000
31.464
9.966
991
3113.32
771326.41
982,081
973,242,271
31.480
9.970
992
3116.46
772883.86
984,064
976,191,488
31.496
9.973
993
3119.60
774442.88
986,049
979,146,657
31.512
9.977
994
3122.75
776003.47
988,036
982,107,784
31.528
9.980
995
3125.89
777565.63
990,025
985,074,875
31.544
9.983
996
3129.03
779129.36
992,016
988,047,936
31.559
9.987
997
3132.17
780694.66
994,009
991,026,973
31.575
9.990
998
313511
782261.54
996,004
994,011,992
31591
9993
999
3138.45
783829.98
998,001
997,002,999
31.607
9.997
1000
3141.60
785400.00
1,000,000
1,000,000,000
31.623
10.000
CIRCLES: — DIAMETER, CIRCUMFERENCE, &C.
87
TABLE No. IV. CIRCLES:— DIAMETER, CIRCUMFERENCE,
AREA, AND SIDE OF EQUAL SQUARE.
Diamecer.
Grcum
fereoce.
Area.
Side of
Equal Souare
(Square Root
of Area).
Diameter.
Circum
ference.
Area.
Side of
Equal Square
(Square Root
of Area).
S/«6
H
1 t
9/16
) H
'J"
.1963
.3927
5890
.7854
.9817
I.I781
. 1.3744
1.5708
1.7771
1.9635
2.1598
2.3562
2.5525
2.7489
2.9452
.00307
.01227
.02761
.04909
.07670
.1104
.1503
.1963
.2485
.3068
.3712
.4417
.5185
.6013
.6903
.0553
.1107
.1661
.2215
.2770
.3323
.3877
.4431
.4984
.5539
.6092
.6646
.7200
.7754
.8308
3 ,
3'A
3 3/.«
3X
'J'
37/16
3 9A6
3H
It
3'5/i6
9.4248
9.62 II
9.8175
10.014
10.210
10.406
10.602
10.799
10.995
II. 191
11.388
11.584
II.781
11.977
12.173
12.369
7.0686
7.3662
7.6699
7.9798
8.2957
8.6180
8.9462
9.2807
9.6211
9.9680
ia320
10.679
11.044
II.4I6
11.793
12,177
2.6586
2.7140
2.7694
2.8248
2.8801
2.9355
2.9909
3.0463
3.1017
3.1571
3.2124
32678
33232
3.3786
3.4340
3.4894
I
I 5/:6
i 9/16
I"Vr6
3.1416
3.3379
3.5343
37306
3.9270
4.1233
4.3197
4.5160
47124
49087
5. 105 1
5.3014
54978
S.694I
t.8905
60868
.7854
.8866
.9940
1.1075
1.227 1
1.3530
1.4848
1.6229
1.767 1
1.9175
2.0739
2.2365
2.4052
2.5800
2.761 1
2.9483
.8862
.9416
.9969
1.0524
I.IOI7
I.163I
I.2185
1.2739
1.3293
1.3847
1. 4401
1.4955
1.5508
1.6062
I.6616
I.7170
43/16
4X
4 7/,6
49/16
4'S/i6
12.566
12.762
12.959
13155
13351
13547
13744
13.940
14.137
14.333
14.529
14.725
14.922
15.119
15.315
15.511
12.566
12.962
13364
13772
14186
14606
15.033
15465
15.904
16.349
16.800
17.257
17.720
18.190
18.665
19.147
3.5448
3.6002
3.6555
37109
3.7663
3.8217
3.8771
39325
3.9880
4.0434
4.0987
4.1541
4.2095
42648
43202
43756
2
2 V16
2 3/,6
2 5/,6
! 2^
2 7/.6
2>^
2 9/x6
2;i
2»»yi6
2'3/,6
2«
2'5/«6 1
6.2832
6^795
6.6759
6.J722
7.0686
72*49
74*13
7.6576
7.8540
8.0533
8.2^67
8.4430
8.6314
8.83f7
9.0311
3.1416
3.3380
3.5465
3.7584
3.9760
4.2000
4.4302
3.7066
4.9087
51573
54119
5.6723
59395
6.2126
6.4918
6.7772
1.7724
1.8278
1.8831
1.9385
1.9939
2.0493
2.1047
2. 1 601
2.2155
2.2709
2.3262
2.3816
2.4370
2.4924
2.5478
2.6032
5
5 '/16
5 3/,6
59/16
St
5'5/i6
15.708
15.904
16.100
16.296
16493
16.689
16.886
17.082
17.278
17.474
17671
17.867
18.064
18.261
18.457
18.653
19.635
20.129
20.629
21.135
21.647
22.166
22.690
23.221
23.758
24301
24850
25.406
25.967
26.535
27.108
27.688
4.4310
44864
45417
45971
46525
4.7079
4.7633
48187
4.8741
49295
49848
5.0402
50956
5.1510
5.2064
5.2618
MATHEMATICAL TABLES.
CIRCLES: — DIAMETER, CIRCUMFERENCE, &C.
89
Side of
Side of
.Diameter.
Circum
ference.
Area.
Equal Sauare
(Square Root
of Area).
Diameter.
Circum
ference.
Area.
Equal Sauare
(Square Root
of Area}.
iS
56.548
254.469
15951
24
75398
452.390
21.268
liyi
56.941
258.016
16.062
24}i
75791
457115
21.379
i&%
57.334
261.587
16.173
24X
76.183
461.864
21 490
tSH
57.726
265.182
16.283
24^
76.576
466.638
21.601
18^
58.119
268.803
16.394
24K
76.969
471.436
21.712
18H
58.512
272.447
16.505
24H
77.361
476.259
21.822
ii^
58.905
276.117
16.616
24«'
777SA
481.106
21.933
i»H
59.297
279.811
16.727
24;^
78.147
485.978
22.044
19
59.690
283.529
16.837
25
78.540
490.875
22.155
19X
60.083
287.272
16.948
25 >i
78.932
495796
22.265
19V
60.475
291.039
17.060
25 X
79325
500.741
22.376
t9H
6a868
294.831
17.170
25^
79718
505.711
22.487
•9>i
61.261
298.648
17.280
25 K
80.110
510.706
22.598
19^
61.653
302.489
17.391
2SH
80.503
515725
22.709
19^
62.046
306.355
17.502
2sH
80.896
520.769
22.819
i9;<
62.439
310.245
17.613
25^
81.288
525.837
22.930
1 
20
62.832
314.160
17.724
26
81.681
530.930
23.041
20X
63.224
318.099
17.834
26%
82.074
536.047
23.152
20X
63.617
322.063
17.945
26X
82.467
541.189
23.062
20^
64.010
326.05 I
18.056
26^
82.859
546.356
23.373
70}4
64.402
330.064
18.167
26X
83.252
551547
23484
20>i
64795
334101
18.277
26^
83.645
556.762
23.595
204i^
65.188
338.163
18.388
26X
84.037
562.002
23.708
20;<
65.580
342.250
18.499
26^
84.430
567.267
23.816
21
65.973
346.361
18.610
27
84823
572.556
23.927
21^
66.366
350.497
18.721
27%
85.215
577.870
24.038
21X
66.759
354.657
18.831
27H
85.608
583.208
24.149
21H
67.151
358.841
18.942
27H
80.001
588.571
24.259
21X
67.544
363.05 1
19053
27 K
86.394
593.958
24.370
21^
67.937
367.284
19.164
27H
86.786
599370
24481
. 21^
68.329
371.543
19.274
27H
87.179
604.807
24.592
. 21^
68.722
375.826
19.385
27%
87.572
610.268
24.703
1
22
69.115
380.133
19.496
28
87.964
615753
24.813
22X
69.507
384465
19.607
28^
88.357
621.263
24924
22 V
69.900
388.822
19.718
28X
88.750
626.798
25.035
22^
70.293
393.203
19.828
28^
89.142
632.357
25.146
5 22>i
70.686
397.608
19.939
28X
89.535
637.941
25.256
22K
ji.orj^
402.038
20.050
28;^
89.928
643.594
25.367
i^H
7^471
406.493
20.161
28^
90.321
649.182
25.478
22 J<
71.864
410.972
20.271
28?^
90.713
654839
25.589
23
72.256
415.476
20.382
29
91.106
660.521
25.699
23^
72.649
420.004
20.493
29^
91.499
666.227
25.810
23X
73.042
424557
20.604
29X
91.891
671.958
25.921
23>i
73434
429.135
20.715
29^
92.284
677.714
26.032 ,
23;^
73.827
433.731
20.825
2rA
92.677
683.494
26.143
23H
74.220
438.363
20.936
29H
93.069
689298
26.253 .
23JK'
74.613
443.014
21.047
293^
93462
695.128
26.364
23^
75.005
447.699
21.158
29;^
93855
700.981
26.478
rX
MATHEMATICAL TABLES.
Sid= or
Side of
Dinmrtcr,
^—
a™.
Equal Smian
iSquirrlt™.
.
a™.
iSquilTRool
Of Ana).
30
94.248
706.860
26.586
^^.
113.097
10.7.87
31903
30 Ji
94.640
712.762
26.696
36 J^
113.490
1024.9s
32.014
30^
95033
718.690
26.807
36X
113883
1032.06
32.124
30«
95.426
724.641
26.918
36«
114.27;
103919
32235
3o;i
95818
730.618
27.029
36X
1 14.668
1046.3 s
32349
3o«
96.211
736.619
27.139
i(>H
115.061
■05352
1060.73
32.457
742.644
27.250
3^X
115.453
32567
748.694
27.361
36^
MS.846
1067.95
32.678
7'; 4.760
27.472
37
1 16.239
1075.21
32.789
760.868
27.583
37 fi
1 [6,631
1082.48
32,900
766.992
27.693
37X
117,024
1089.79
33011
773.140
27,804
37H
117.417
1097.11
33.021
779'3i3
27.915
y}%
117,810
1104.46
33.232
78;.sio
28.026
37H
1U..84
33343
791.732
28. 136
yjH
[•.l$l
1 1 [9.24
33.454
797.978
28.247
37Pi
1126.66
33564
804.249
28.358
3^ .
119,380
1.34.11
33.675
810.545
28.469
38 H
119773
1.41.59
33786
816.865
28.580
38y
120.166
33.897
823.209
28.69.
3m
120,558
34.008
829.578
28.801
38 Ji
120,951
34118
835972
28.9.2
iSH
121.344
117173
34229
842.390
29.023
3SH
121.737
1179.32
34.340
848.833
29.133
3i^
122.1Z9
1.86.94
3445"
855.30
29.244
39
122,522
119459
34.561
861.79
^9.355
39>i
122,9.5
1202.26
34.672
868.30
29,466
39X
123.307
1209.95
34.783
874.84
^9577
39^
123.700
1217.67
34.894
881.41
29,687
39^
124.093
124485
1225.42
35005 J
888.00
29,798
39H
123318
35115
894.61
29.909
39>^
.24.878
! 240.98
35226 '
901.25
30.020
39H
125.271
1248,79
35337 1
007.92
3o>3'
40
125.664
.256.64
35448
914.61
30,241
40H
126.056
1264.50
3SSSS
921.32
30.352
40V
126.449
1272.39
35.669
928.06
30.463
^oH
126.842
.280.31
35780
93482
30574
Ao'A
127.334
1288.25
35891
941.60
30.684
AoH
127.627
1296.21
36.002
948.41
30.79s
40K
128.020
1304.20
36.112
95525
30,906
4o?i
128.412
13.2.2I
36,223
962.11
31.017
41
128.805
1320.25
36.334.
968.99
31.128
4i>i
129.198
1328.32
36.445
975.90
31238
41X
129.591
129.983
1336.40
36.555
982.84
31.349
41?^
13445'
36.666
989.80
31.460
41^
130.376
1352.65
36.777
996.78
31571
4'X
130.769
1360.81
36,888
1003,78
31.681
41 Ji"
131. .61
1369.00
36.999
1010.82
31792
At 'A
13'SS4
137721
37109
circles: — DIAMETER, CIRCUMFERENCE, &C.
91
DniBeter.
Grcum
ference.
Area.
Side of
Equal Sauare
(Square Root
of Area).
Diameter.
Circum
ference.
Area.
Side of
Equal Square
(Square Root
of Area).
1
42
42'A
42H
4J^
131947
132.339
132.732
133.125
133.518
i33.9'o
134.303
134.696
1385.44
139370
1401.98
1410.29
14x8.62
1426.98
143536
144377
37.220
37.331
37.442
37.552
37.663
37.774
37.885
37.996
48
48 >^
48X
48>i
48^
1 50.796
151.189
151.582
151974
152.367
152.760
153.153
153.545
1809.56
1818.99
1828.46
1837.93
1847.45
1856.99
1866.55
1876.13
42.537
42.648
42.759
42.870
42.980
43.091
43.202
43.313
•« !
;43H i
j43V ;
43H
43><
.43M
43V
•43J<
135.088
135.481
, 135.874
1 136.266
136.659
137.052
137.445
137.837
1452.20
1460.65
1469.13
147763
1486.17
1494.72
1503.30
I5II.90
38.106
38.217
38.328
38.439
38.549
38.660
38.771
38.882
49 ^
49V
49H
49>i
A9H
49H
49?^
153.938
154331
154723
155.116
155.509
155.901
156.294
1 56.687
1885.74
1895.37
1905.03
1914.70
1924.42
193415
1943.91
195369
43.423
43.534
43.645
43.756
43867
43.977
44.088
44.199
'44
44J<
UH
uH ,
138.230
138.623
139.015
139.408
139.801
140.193
140.586
. 140.979
1520.53
1529.18
1537.86
1546.55
1555.28
1564.03
1572.81
1581.61
38.993
39.103
39.214
39.325
39.436
39.546
39657
39.768
50
50V
1 57.080
157.865
158.650
1 59.436
1963.50
1983.18
2002.96
2022.84
44.310
44.531
44.753
44.974
160.221
161.007
161.792
162.577
2042.82
2062.90
2083.07
2103.35
45196
45417
45639
45.861
4> 1
141.372
141.764
142.157
142.550
142.942
143.335
143.728
144.120
1590.43
1599.28
1608.15
1617.04
1625.97
1634.92
1643.89
1652.88
39.879
39.989
40.110
40.2 1 1
40.322
40.432
40.543
40.654
45H
■45X
AiH
45K
45^
45V
52 ,
52X
163.363
164.148
164.934
165.719
2123.72
2144.19
2164.75
2185.42
46.082
46.304
46.525
46.747
53 ,
S3X
S3>i
S3J<
166.504
167.290
168.075
168.861
2206.18
2227.05
2248.01
2269.06
46.968
47.190
47.411
47.633
46
46H
46X
46X
46>i
46M
46,V
46J<
144.513
144906
145.299
145.691
146.084
146.477
146.869
147.262
1661.90
1670.95
1680.01
1689.10
1698.23
1707.37
1716.54
1725.73
40.765
40.876
40.986
41.097
41.208
41.319
41.429
41.540
54 ,
S4X
S4>^
54JC
169.646
170.431
I7I.217
172.002
2290.22
2311.48
2332.83
2354.28
47.854
48.076
48.298
48.519
SS
55K
S5^
172.788
173573
174.358
175144
2375.83
239748
2419.22
2441.07
48.741
48.962
49.184
49405
47
47H
'47V
, 47«
,47>i
i47M :
i47V !
,47J< 1
» ____
147.655
148.047
148.440
148.833
149.226
149.618
1 50.01 1
150.404
173494
1744.18
175345
1762.73
1772.05
1781.39
1790.76
1800.14
41.651
41.762
41.873
41.983
42.094
42.205
42.316
42.427
56
s(>'A
175.929
176.715
177.500
178.285
2463.01
2485.05
2507.19
2529.42
49.627
49.848 •
50.070
50.291
92
MATHEMATICAL TABLES.
Diameter.
Circum
ference.
Area.
Side of
Equal Square
l( Square Root
of Area;.
Diameter.
Circum
ference.
Area.
Side of
Equal Souare
(Square Root
of Area).
57 ^
57X
57>i
57^
179.071
179.856
180.642
181.427
2551.76
2574.19
2596.72
2619.35
50.513
50.735
50.956
51.178
68
68X
68^
68^
213.628
214.414
215.199
215.985
3631.68
3658UH
3685.29
3712.24
60.261
60.483
60.704
60.926
58
58X
58^
182.212
182.998
183.783
184.569
2642.08
2664.91
2687.83
2710.85
51.399
51.621
51.842
52.064
69
69X
69X
69^
216.770
217.555
218.341
219.126
3739.28
3766.43
3793.67
3821.02
61.147
61.369
61.591
61.812
59 ,
59X
59>i
59^
185.354
186.139
186.925
187.710
2733.97
2757.19
2780.51
2803,92
52.285
52.507
52.729
52.950
70
70X
70X
219.912
220.697
221.482
222.268
3848.46
3875.99
3903.63
3931.36
62.034
62.255
62.477
62.698
60
60X
60X
6o<'
188.496
189.281
190.066
190.852
2827.44
2851.05
2874.76
2898.56
53.172
53.393
53.615
53.836
71
71X
71^
223.053
223.839
224.624
225.409
3959.20
3987.13
4015.16
4043.28
62.920
63.141
63.363
63.545
61
61X
6i>i
6i?<
191.637
192.423
193.208
193.993
2922.47
2946.47
2970.57
2994.77
54.048
54.279
54.501
54.723
72
72^
72 >i
72X
226.195
226.980
227.766
228.551
1
4071.51
4099.83
4128.25
4156.77
63.806
64.028
64.249
64.471
62
62X
62 >i
62«'
194.779
195.564
196.350
197.135
3019.07
3043.47
3067.96
3092.56
54.944
55.166
55.387
55.609
73
73X
7Z)i
229.336
230.122
230.907
231.693
4185.39
42 14. II
4242.92
4271.83
64.692
64.914
65.135
65.357
63
63^
63>i
63^
197.920
198.706
199.491
200.277
3117.25
3142.04
3166.92
319I.9I
55.830
56.052
56.273
56.495
74
74V
U%
74H
232.478
233.263
234.049
234.834
4300.85
4329.95
4359.16
4388.47
65.578
65.800
66.022
66.243
64
64X
64>i
64l<'
201.062
201.847
202.633
203.418
3216.99
3242.17
3267.46
3292.83
56.716
56.938
57.159
57.381
75 ,
7SX
75>i
7S)i
235.620
236.405
237.190
237.976
4417.87
4447.37
4476.97
4506.67
66u^65
66.686
66.908
67.129
65
6SX
65 >4
6SJ<
204.204
204.989
205.774
206.560
3318.31
3343.88
3369.56
3395.33
57.603
57.824
58.046
58.267
76
763^
76;^
76m:
238.761
239.547
240.332
241. 117
4536.47
4566.36
4596.35
4626.44
67.351 1
67.572 1
67.794
68.016
66
66X
66;^
66m:
207.345
208.131
208.916
209.701
3421.20
3447.16
3473.23
3499.39
58.489
58.710
58.932
59.154
77
77X
77^
773i
241.903
242.688
243.474
244.259
4656.63
4686.92
4717.30
4747.79
68.237
68.459
68.680
68.902
67X
67X
67X
210.487
211.272
' 212.058
212.843
1
3525.66
3552.01
3578.47
3605.03
59.375
59.597
59.818
60.040
78
78X
78X
7»H
245.044
245.830
246.615
247.401
4778.37
4809.05
4839.83
4870.70
69.123
69.345
69.566
69.788
CIRCLES: — DIAMETER, CIRCUMFERENCE, &C
93
Dnuneter.
Circuia
fcrence.
Area.
Side of
Equal Souare
(Square Root
of Area).
Diameter.
Circum
ference.
Area.
Side of
Equal Souare
(Square Root
of Area).
79.'^
79«'
248.186
248.971
249.757
, 250.542
4901.68
4932.75
4963.92
4995.19
70.009
70.231
70.453
70.674
90
90X
90X
9°}i
282.744
283.529
284.314
285.099
6361.74
6399.12
6432.62
6468.16
79.758
79.980
80.201
80.423
80
80V
80V
251.328
252.113
252.898
253.683
5026.56
5058.00
5089.58
5121.22
70.896
7I.II8
71.339
71.561
91
91X
9i>i
9«v:
285.885
286.670
287.456
288.242
6503.89
653968
6573.56
6611.52
80.644
80.866
81.087
81.308
81
8iy
8i>i
81 V
254.469
255.254
256.040
256.825
5153.00
5184.84
5216.82
5248.84
71.782
72.004
72.225
72.447
92
92X
92H
289.027
289.812
290.598
291.383
6647.62
6683.80
6720.07
6756.40
81.530
81.752
81.973
82.195
82
83V
82>i
82V
1 257.611
258.396
259.182
, 259.967
5281.02
5313.28
534562
5378.04
72.668
72.890
73.111
73.333
93 ^
93^
93X
93V
292.168
292.953
293.739
294.524
6792.92
6829.48
6866.16
6882.92
82.416
82.638
82.859
83.081
■83
J83V
83}^
83V
< 260.752
' 261.537
262.323
263.108
5410.62
5443.24
5476.00
5508.84
73.554
73.776
73.997
74.219
94,
94V
94^
94V
295.310
296.095
296.881
297.666
6939.79
6976.72
7013.81
7050.92
83.302
83.524
83.746
83.968
84
84,V
84^
84V
1 263.894
264.679
265.465
266.250
5541.78
5574.80
5607.95
5641.16
74.440
74.662
74.884
75.106
95 ,
95X
95 >^
9SH
298.452
299.237
300.022
300.807
7088.23
7125.56
7163.04
7200.56
84189
84.411
84632
84854
85
8;V
85V
267.036
267.821
268.606
269.392
5674.51
5707.92
5741.47
5775.09
75.327
75.549
75.770
75.992
96
96V
96'A
96V
301.593
302.378
302.164
303.948
7238.24
7275.96
7313.84
735172
85.077
85.299
85.520
85.742
186
86V
86>i
86V
270.177
270.962
271.748
272.533
5808.81
5842.60
5876.55
5910.52
76.213
76.435
76.656
76.878
97
97V
97 >i
97^
304.734
305.520
306.306
307.090
7389.80
7427.96
7474.20
7504.52
85.963
86.185
86.407
86.628
87
87V '
87>i !
87V ;
273.319
274104
274890
275.675
5944.69
5978.88
6013.21
6047.60
77.099
77.321
77.542
77.764
98
98X
98^
307.876
308.662
309.446
310.232
7542.96
7581.48
7620.12
7658.80
86.850
87.072
87.293
87.515
88
88V
88 Ji
88V
276.460
277.245
278.031
278.816
6082.13
6116.72
6151.44
6186.20
77.985
78.207
78.428
78.650
99 ,
99^
99'A
99^
3II.OI8
311.802
312.588
313.374
7697.68
7736.60
7775.64
7814.76
87.736
87.958
88.180
88.401
100
100^
314.159
315730
7854.00
7932.72
88.623
89.066
59
89V
89V
279.602
280.387
281.173
281.958
6221.15
6256.12
6291.25
6326.44
78.871
79.093
79.315
79.537
lOI
317.301
318.872
8011.84
8091.36
89.509
89.952
94
MATHEMATICAL TABLES.
Diameter.
Circum
ference.
Area.
Side of
Equal Square
^Square Root
of Area).
Diameter.
Circum
ference.
Area.
Side of
Equal Souare
(Square Root
of Area).
1 02
I02X
320.442
322.014
8171.28
8251.60
90.395
90.838
112
II2X
351.858
353.430
9852.03
9940.20
99.258
99.701
103
IO3X
323.584
325.154
8332.29
8413.40
91.282
91.725
113
II3>^
355.000
356.570
10028.75
loi 17.68
100.144
100.587
104
io4}4
326.726
328.296
8494.87
8576.76
92.168
92.61 1
114
358,142
359.712
10207.03
10296.76
101.03 1
101.474
105
329.867
331.438
8659.01
8741.68
93.054
93.497
115 .
"5>^
361.283
362.854
10386.89
10477.40
IOI.917
102.360
106
333.009
334.580
8824,73
8908.20
93.940
94.383
116
116;^
364.425
365.996
10568.32
10659.64
102.803
103.247
107
107 >^
336.150
337.722
8992.02
9076.24
94.826
95.269
117
367.566
369.138
10751.32
10843.40
103.690
104.133
108
108K
339.292
340.862
9160.88
9245.92
96.156
118
ii8>^ ,
370.708
372.278
10935.88
1 1028.76
104.576
105.019
109
109;^
342.434
344.004
9331.32
9417.12
96.599
97.042
119
119^
373.849
375.420
1 1 122.02
1 121 5.68
105.463
105.906
no
345.575
347.146
9503.32
9589.92
97.485
97.928
120
376.991
M 309.73
106.350
III
MiX
348.717
350.288
9676.89
9764.28
98.371
98.815
LENGTHS OF CIRCULAR ARCS.
95
TABLE No. v.— LENGTHS OF CIRCULAR ARCS FROM
l^ TO l8o^ GIVEN, THE DEGREES.
(Radius = i.)
Degrees.
Length.
' Degrees.
1
Length.
Degrees.
Length.
Degrees.
Length.
I
.0174
40
.6981
79
1.3788
117
2.0420
2
.0349
41
.7156
80
13963
118
2.0595
3
0524
42
.7330
119
2.0769
4
.0698
43
.7505
81
14137
5
.0873
44
.7679
82
I.4312
120
2.0944
6
.0147
45
.7854
83
1.4486
121
2.II18
7
.0222
46
.8028
84
1. 4661
122
2.1293
8
.0396
47
.8203
85
1.4835
123
2.1467
9
0571
48
.8377
86
I.50IO
124
2.1642
49
.8552
87
1.5^84
125
2.I8I7
lO
.1745
^ •
w %/
88
15359
126
2.1991
II
.1920
50
.8727
89
15533
127
2.2166
12
.2094
SI
.8901
^%
128
2.2304
13
.2269
5»
.9076
90
1.5708
129
2.2515
14
.2443
S3
.9250
91
L5882
15
.2618
54
.9424
92
1.6057
130
2.2689
16
.2792
SS
•9599
93
L6231
131
2.2864
17
.2967
S6
•9774
94
1.6406
132
2.3038
18
314I
57
.9948
95
L6581
133
2.3213
19
'ZZ^^
S8
1.0123
96
1.6755
134
2.3387
^
59
1.0297
97
1.6930
135
2.3562
ao
.3491
98
L7I04
136
2.3736
21
.3665
60
1.0472
99
1.7279
137
2.391 1
22
.3840
61
1.0646
138
2.4085
23
.4014
62
1. 0821
100
1.7453
139
2.4260
24
.4189
63
10995
lot
1. 2628
1
25
•4363
64
1.1170
102
L7802
140
2.4435
26
.4538
65
II345
103
1.7977
141
2.4609
27
.4712
66
1.1519
104
1.8151
142
2.4784
28
.4887
67
1. 1694
105
1.8326
143
2.4958
29
.5061
68
1. 1868
106
1.8500
144
2.5133
^
69
1.2043
107
1.8675
145
2.5307
30
.5236
108
L8849
C46
2.5482
31
.54IC)
70
1. 2217
109
1.9024
147
2.5656
32
.5585
71
1.2392
148
2.5831
33
.5759
72
1.2566
no
1.9199
149
2.6005
34
.5934
73
12741
III
19373
. 35
.6109 ,
74
L2915
112
1.9548
150
2.6180
36
.6283
75
1.3090
113
1.9722
151
2.6354
37
.645S !
76
1.3264
114
1.9897
152
2.6529
38
.66:^2
77
1.3439
115
2.0071
153
2.6703
1 39
.6807
78
1.3613
ii6
2.0246
154
2.6878
i:"^i.
96
MATHEMATICAL TABLES.
Degrees.
Length.
Degrees.
Length.
Degrees.
Length.
Degrees.
Length.
155
2.7053
161
2.8100
168
2.9321
174
3.0369
156
2.7227
162
2.8274
169
2.9496
175
30543
157
2.7402
163
2.8449
176
3.0718
158
2.7576
164
2.8623
170
2.9670
177
3.0892
159
2.7751
165
2.8798
171
2.9845
178
3.1067
166
2.8972
172
3.0020
179
3. 1 241
160
2.7925
167
2.9147
173
30194
180
3.I416
L
LENGTHS OP CIRCULAR ARCS.
97
TABLE No. VI.— LENGTHS OF CIRCULAR ARCS, UP TO A
SEMICIRCLE. GIVEN, THE HEIGHT.
(Chord = I.)
Height.
Length.
HdghL
Length.
Height.
Length.
Height.
Length.
.100
1.02645
.140
I.05147
.180
1.08428
.220
1. 1 2445
.101
1.02698
.141
1.05220
.181
I.08519
.221
I.I2556
I02
1.02752
.142
1.05293
.182
I.0861I
.222
1. 12663
.103
1.02806
.143
1.05367
.183
1.08704
.223
1.12774
.104
1.02860
.144
I.0544I
.184
1.08797
.224
1. 12885
105
1. 02914
.145
I.05516
.185
1.08890
.225
1. 12997
.106
1.02970
.146
I.0559I
.186
1.08984
.226
1.13108
.107
1.03026
.147
1.05667
.187
1.09079
.227
I.I3219
.108
1.03082
.148
1.05743
.188
I.O9174
.228
1.13331
.109
103139
.149
I.05819
.189
1.09269
.229
1. 13444
.110
I.O3I96
.150
1.05896
.190
1.09365
.230
I.I3557
.III
1.03254
.151
1.05973
.191
I.O9461
.231
I.I3671
.112
I.O3312
.152
1.0605 1
.192
1.09557
.232
I.I3786
."3
I.O3371
.153
1. 06130
.193
1.09654
.233
113903
.114
1.03430
.154
1.06209
.194
1.09752
•234
1. 14020
.1^5
1.03490
•155
1.06288
.195
1.09850
.235
I.I4136
.116
I.O3551
.156
1.06368
.196
1.09949
.236
I.I4247
.117
I.O361I
.157
1.06449
.197
1. 10048
.237
114363
.118
1.03672
.158
1.06530
.198
I.IOI47
.238
1. 14480
.119
1.03734
.159
I.0661I
.199
1. 10247
.239
r. 14597
.120
1.03797
.160
1.06693
.200
1. 10348
.240
1.14714
.121
1.03860
.161
1.06775
.201
1. 10447
.241
114831
.122
1.03923
.162
1.06858
.202
I.IO548
.242
1. 14949
.123
1.03987
.163
I.0694I
.203
1. 10650
.243
1. 15067
.124
1.0405 1
.164
1.07025
.204
I.IO752
.244
1.15186
•125
I.04I16
.165
1. 07109
.205
I.I0855
.245
1.15308
.126
I.04181
.166
I.07194
.206
I.IO958
.246
1.15429
.127
1.04247
.167
1.07279
.207
I.II062
.247
1.15549
.128
1.043 13
.168
1.07365
.208
I.II165
.248
1.15670
.129
1.04380
.169
I.07451
.209
I.II269
.249
1.15791
.130
1.04447
.170
1.07537
.210
I.II374
.250
1.15912
131
I.O4515
.171
1.07624
.211
I.II479
.251
1. 16033
.132
1.04584
.172
I.O771I
.212
I.II584
.252
1.16157
'^33
1.04652
.173
1.07799
.213
I.I1692
•253
1. 16279
•134
1.04722
.174
1.07888
.214
I.II796
.254
1. 16402
135
1,04792
.175
1.07977
.215
1. 1 1904
.255
1. 16526
.136
1.04862
.176
1.08066
.216
I.I20II
.256
1. 16649
•^37
1.04932
.177
I.08156
.217
I.I2II8
.257
1.16774
.13S
•^39 /
1.05003
.178
1.08246
.218
1. 12225
.258
1. 16899
105075 \
.179
1.08337
.219
I.I2334
.259
1. 17024
98
MATHEMATICAL TABLES.
Height,
Length.
.260
.261
.262
.263
.264
.265
.266
.267
.268
.269
.270
.271
.272
.273
.274
.275
.276
.277
.278
.279
.280
.281
.282
.283
.284
.285
.286
.287
.288
.289
.290
.291
.292
•293
.294
•295
.296
.297
.298
.299
.300
.301
.302
•303
•304
•305
.306
.17150
.17275
.17401
.17527
.17655
.17784
.17912
.18040
.18162
.18294
.18428
•18557
.18688
.18819
.18969
.19682
.19214
.19345
•19477
.19610
.19743
.19887
.20011
.20146
.20282
.20419
.20558
.20696
.20828
.20967
.21202
.21239
.21381
.21520
.21658
.21794
.21926
.22061
.22203
.22347
.22495
.22635
.22776
.22918
.23061
.23205
•23349
Height
Length.
.307
.308
.309
.310
.311
.312
.313
.314
.315
.316
.317
.318
.319
.320
.321
.322
.323
.324
.325
.326
•327
.328
•329
.330
•332
'333
.334
.335
•336
.337
.338
.339
.340
•341
.342
.343
.344
•345
.346
•347
.348
.349
.350
.351
.352
.353
•23494
.23636
.27780
•23925
.24070
.24216
.24360
.24506
.24654
.24801
.24946
.25095
.25243
.25391
•25539
.25686
.25836
.25987
•26137
.26286
.26437
.26588
.26740
.26892
.27044
.27196
.27349
.27502
.27656
.27810
.27864
.28118
.28273
.28428
.28583
.28739
.28895
.29052
.29209
.29366
•29523
.29681
•29839
.29997
.30156
.30315
.30474
Height.
•354
•355
•356
•357
•358
.359
.360
.361
.362
'3^3
.364
.365
.366
.367
.368
.369
.370
.371
.372
.373
.374
.375
.376
.377
.378
.379
.380
.381
.382
.383
.384
.385
.386
.387
.388
.389
.390
.391
.392
•393
•394
.395
.396
.397
.398
•399
.400
Length.
•30634
•30794
•30954
.31115
.31276
•31437
•31599
.31761
•3^923
.32086
•32249
.32413
•32577
.32741
•32905
.33069
.33234
.33399
.33564
.33730
.33896
.34063
.34229
.34396
.34563
.34731
.34899
.35068
•35237
.35406
•35575
•35744
.35914
.36084
.36254
.36425
.36596
.36767
•36939
.37111
.37283
.37455
.37628
.37801
.37974
.38148
.38322
Height.
.401
.402
.403
.404
.405
.406
.407
.408
.409
.410
.411
.412
.413
.414
.415
.416
.417
.418
.419
.420
.421
.422
•423
.424
.425
.426
.427
.428
.429
.430
.431
.432
.433
.434
.435
.436
•437
.438
•439
.440
.441
.442
.443
.444
•445
.446
.447
Length.
[.38496
.38671
.38846
•39021
.39196
39372
•30548
•39724
.39900
.40077
.40254
.40432
.40610
.40788
.409^6
.41145
■41324
■41503
.41682
.41861
.42041
.42222
.42402
•42583
.42764
•42945
•43127
•43309
.43491
•43673
•43856
•44039
.44222
.44405
.44589
•44773
•44957
■45142
■45327
•455"
•45697
■45883
.46069
•46255
.46441
.46628
.46815
LENGTHS OF CIRCULAR ARCS.
99
Hdght.
.448
•449
.450
.451
•452
•453
•454
•455
•45 <^
•457
.458
•459
JLength.
1.47002
1. 47189
1.47377
1.47565
1.47753
1.47942
I.48131
1.48320
1.48509
1.48699
1.48889
1.49079
.460 1.49269
Height.
.461
.462
.463
.464
.465
.466
.467
.468
.469
.470
.471
.472
.473
.474
Length.
.49460
.49651
.49842
•50033
.50224
.50416
.50608
.50800
.50992
•51185
•51378
■5157I
.51764
•51958
Height
Length.
.475
.476
.477
.478
.479
480
.481
.482
.483
.484
.485
.486
.487
.488
52^52
52346
52541
52736
52931
53126
53322
53518
53714
53910
54106
54302
54499
54696
Height
.489
.490
.491
.492
•493
•494
•495
496
.497
.498
499
.500
Length.
154893
55090
.55288
.55486
.55685
•55854
.56083
.56282
.56481
.56680
.56879
57079
MATHEMATICAL TABLES.
TABLE No. VIL— AREAS OF CIRCULAR SEGMENTS, UP TO A
SEMICIRCLE.
AREAS OF CIRCULAR SEGMENTS.
101
, HdghL
1
1
Area.
Height.
Area.
Height.
Area.
Height.
1
1
Area.
'.X57
.07892
.203
.11423
.249 ;
.15268
•295
.19360
.158
.07965 '
.204
.11504
.250
• IS3SS
.296
. 19451
•159
.08038
.205
.11584
•251
.252
.15442
.15528
.297
•19543
.160
.08111
.206
.11665
.298
.19634
.161
.162
.08185
.08258
.207
.208
>k ^v ^^
.11746
.11827
.11908
.253
.254
.15615
.15702
.299
.300
•19725
.19817
.163
.08332
.209
.255
.15789
.301
.19908
.164
.08406
.210
.11990
.256
.15876
.302
.20000
.165
.08480
.211
I2071
.257
.15964
•303
.20092
. .166
.08554
.212
I2153
.258
.16051
.304
.20184
.167
.08629
.213
12235
•259
.16139
•305
.20276
.168
1 ^
.08704
.214
I2317
.260
.16226
.306
.20368
.169
.08778
.215 .
.12399
.261
.16314
•307
.20460
.170
.08854
.216
1 248 1
.262
.16402
.308
•20553
.171
.08929
.217
12563
.263
.16490
•309
.20645
.172
.09004
.218
12646
.264
.16578
.310
.20738
.173
.09080
.219
12729
.265
16666
%0
•3"
.20830
.174
•09155
.220
I281I
.266
1675s
.312
.20923
•175
.09231
.221
12894
.267
16843
•313
.21015
.176
.09307
.222
12977
.268
16932
•314
.21108
•'^2
09383
.223
13060
.269
17020
•315
.21201
.178
.09460
.224
I3144
.270
.17109
.316
.21294
.179
.09537
.225
13227
.271
.17198
.317
.21387
.180
.09613
.226
133"
.272
.17287
.318
.21480
.181
.09690
.227
13395
.273
.17376
•319
•21573 .
.182
.09767
.228
13478
.274
.17465
.320
.21667
.183
.09845
.229
13562
.275
17554
•J
•321
.21760
.184
.09922
.230
13646
.276
.17644
.322
•21853
i'f5
.09200
.231
I373I
.277
'I7733
•323
.21947
.186
.10077
.232
I3815
.278
.17823
.324
.22040
.187
€%
.10153
.233
13899
.279
.17912
.325
.22134
.188
.10233
.234
13984
.280
.18002
.326
.22228
.189
.10317
.235
14069
.281
.18092
.327
.22322
.190
.10390
.236
I4I54
.282
.18182
.328
.22415
,.191
.10469
.237
14239
.283
.18272
•329
.22509
.192
.10547
.238
14324
.284
.18362
•330
•332
•333
.334
•335
.22603
.22697
.22792
.22886
.22980
.23074
•193
.194
.195
.196
.197
.198
.» ^ _
.10626
.10705
.10784
.10864
.10943
•239 •
.240
.241
.242
.243
14409
14494
14580
14665
14752
.285
.286
.287
.288
.289
.18452
.18542
.18633
.18723
.18814
.11023
.244
•14837
.290
.18905
.336
.23169
1 .199
.11102
.245
14923
.291
.18996
•337
.23263
.200
.11182
.246
.15009
.292
.19086
•338
•23358
.201
.11262
.247
.15096
•293
.19177
•339
•23453
.202
.11343
.248
.15182
•294
.19268
.340
.^3547
SINES, COSINES, &C OF ANGLES.
103
TABLE No. VIIL— SINES, COSINES, TANGENTS, COTANGENTS,
SECANTS, AND COSECANTS OF ANGLES FROM 0° to 90°
Advancing by 10' or onesixth of a degree. (Radius =i.)
1 Angle.
Sine.
Cosecant.
Tangent
Cotangent
Secant
•
Cosine.
0' 0'
.000000
Infinite.
.000000
Infinite.
1. 00000 ]
[.000000
90° 0'
10
.002909
34377516
.002909
343.77371
1. 00000
.999996
50
20
.005818
171.88831
.005818
171.88540
1.00002
.999983
40
1 30
.008727
114.59301
.008727
114.58865
1.00004
.999962
30
40
.011635
85945609
.011636
85.939791
1.00007
.999932
20
50
.014544
68.757360
.014545
68.750087
1. 000 1 1
.999894
10
I
.017452
57.298688
.017455
57.289962
I.OOOI5
.999848
89
10
.020361
49.114062
.020365
49.103881
1.0002 1
.999793
50
20
.023269
42.975713
.023275
42.964077
1.00027
.999729
40
1 30
.026177
38.201550
.026186
38.188459
1.00034
•999657
30
40
.029085
34.382316
.029097
34.367771
1.00042
.999577
20
' 50
.031992
31.257577
.032009
31.241577
1.0005 1
.999488
10
2
.034899
28.653708
.034921
28.636253
1.0006 1
.999391
88
' 10
.037806
26.450510
•037834
26.431600
1.00072
.999285
50
20
.040713
24.562123
.040747
24.541758
1.00083
.999171
40
30
.043619
22.925586
.043661
22.903766
1.00095
.999048
30
40
.046525
21.493676
.046576
21.470401
1. 00108
.998917
20
SO
.049431
20.230284
.049491
20.205553
1. 00122
.998778
10
3 P
.052336
19.107323
.052408
19.081 137
I.OOI37
.998630
87
10
•055241
18.102619
.055325
18.074977
I.OOI53
.998473
50
20
.058145
17.198434
.058243
17169337
1. 00169
.998308
40
30
.061049
16.380408
.061163
16.349855
1. 00187
•998135
30
40
.063952
15.636793
.064083
15.604784
1.00205
.997357
20
50
.066854
14.957882
.067004
14.924417
1.00224
•997763
10
4
.069756
14.335587
.069927
14.300666
1.00244
.997564
86
10
.072658
i3.763"5
.072851
13.726738
1.00265
.997357
50
20
•075559
13.234717
.075776
13.196888
1.00287
.997141
40
30
.078459
12.745495
.078702
12.706205
1.00309
.996917
30
40
.081359
12.291252
.081629
12.250505
1.00333
.996685
20
50
.084258
11.868370
.084558
II. 826167
1.00357
.996444
10
5
.087156
11473713
.087489
11.430052
1.00382
.996195
85
10
.090053
11.104549
.090421
11.059431
1.00408
.995937
50
20
.092950
10.758488
.093354
10.711913
1.00435
.995671
40
30
.095846
10.433431
.096289
10.385397
1.00463
.995396
30
40
.098741
10.127522
.099226
10.078031
1.0049 1
•995113
20
50
.101635
9.8391227
.102164
9.7881732
1.005 2 1
.994822
10
Angle.
Cosine.
Secant
Cotangent
Tangent
Cosecant
Sme.
I04
MATHEMATICAL TABLES.
Angle.
Sine.
Cosecant.
Tangent.
Cotangent
Secant
Cosine.
6° o'
.104528
9.5667722
.105104
95143645
1.0055 X
•994522
84° 0'
lO
.107421
9.3091699
.X 08046
92553035
X. 00582
.9942x4
50
20
.110313
9.0651512
.X 10990
9.OO9826X
X.O0614
•993897
40
30
.113203
8.8336715
.113936
8.7768874
1.00647
.993572
30
40
.116093
8.613790I
.XX6883
8.5555468
i.oo68x
•993238
20
50
.118982
8.4045586
•119833
8.3449558
1.007x5
.992896
10
7
.121869
8.2055090
.122785
8.x 443464
1.0075X
.992546
83
10
.124756
8.0156450
.125738
7.9530224
X.00787
.992x87
50
20
.127642
78344335
.X 28694
7.7703506
1.00825
.99x820
40
30
.130526
7.6612976
.131653
7.5957541
X. 00863
•991445
30
40
•I334IO
7.4957100
.134613
7.4287064
X.00902
.99106X
20
50
.136292
7^3371909
.137576
7.2687255
X.00942
.990669
10
8
•I39173
7.1852965
.140541
7^ii53697
X. 00983
.990268
82
10
•142053
7.0396220
•143508
6.9682335
X. 0x024
.989859
50
20
.144932
6.8997942
.146478
6.8269437
X. 0x067
•989442
40
30
.147809
6.7654691
.X 4945 1
6.691x562
x.oxxxx
.989016
30
40
.150686
6.6363293
.152426
6.5605538
IOII55
.988582
20
50
•I53561
6^5120812
.155404
6.4348428
1.0x200
.988x39
10
9
.156434
6.3924532
.158384
6.3137515
1.01247
.987688
81
10
•159307
6.2771933
.161368
6.X970279
X. 01294
.987229
50
20
.162178
6.1660674
•164354
6.0844381
1.0x432
.986762
40
30
.165048
6.0588980
.167343
59757644
X.OI39I
.986286
30
40
.167916
59553625
•170334
5.8708042
1.0x440
.98580X
20
50
.170783
5.8553921
.173329
5.7693688
X.OI49I
•985309
XO
10
.173648
57587705
.176327
5.6712818
101543
.984808
80
10
.176512
56653331
.179328
55763786
1^01595
.984298
* 50
20
•179375
55749258
.182332
5.4845052
X. 0x649
.983781
40
30
.182236
5.4874043
•185339
5.3955172
X.OI703
•983255
30
40
.185095
5.4026333
.188359
5^3092793
1.0x758
.98272X
20
50
•187953
5.3204860
.191363
5.2256647
x.ox8x5
.982x78
10
II
.190809
5.2408431
.194380
5.1445540
X. 0x872
.98x627
79
10
.193664
5.1635924
.X9740X
50658352
X. 01930
.981068
50
20
.196517
5.0886284
.200425
4.9894027
1.0x989
.980500
40
30
.199368
5.OI58317
.203452
49151570
1.02049
•979925
30
40
.202218
4.9451687
.206483
4.8430045
1.02110
•979341
20
50
.205065
4.8764907
.209518
4.7728568
1.0217X
.978748
10
12
.207912
4.8097343
.212557
4.7046301
1.02234
.978148
78
10
.210756
4.7448206
•215599
4.6382457
1.02298
•977539
SO
20
•213599
4.6816748
.218645
45736287
1.02362
.976921
40
30
.216440
4.6202263
.22x695
4.5x07085
1.02428
.976296
30
40
.219279
4.5604080
.224748
4.4494x81
1.02494
.975662
20
50
.2221X6
4502x565
.227806
4,3896940
1.02562
.975020
10
1
Coune.
Secant
Cotangent
Tangent
Cosecant
Sine.
Angle.
SINES, COSINES, &C OF ANGLES.
105
Angle.
Sine.
Cosecant.
Tangent.
Cotangent
Secant.
Cosine.
13° 0'
.224951
4.44541 15
.230868
4.3314759
1.02630
•974370
77" 0'
10
.227784
4.3901 158
•233934
4.274706*6
1.02700
.973712
50
20
.230616
4.3362150
.237004
4.2193318
1.02770
•973045
40
30
•233445
42836576
.240079
4.1652998
1.02842
.972370
30
40
.236273
4^2323943
.243158
4.II25614
1. 02914
.971687
20
50
.239098
4,1823785
.246241
4.0610700
1.02987
•970995
10
14
.241922
4.1335655
.249328
4,0107809
I.O3061
.970296
76
1 10
•244743
4.0859130
.252420
3.9616518
IO3137
.969588
50
' 20
•247563
4.0393804
•255517
3.9136420
I.O3213
.968872
40
30
.250380
3.9939292
.258618
3.866713I
1.03290
.968148
30
40
•253195
3.9495224
.261723
3.8208281
1.03363
.967415
20
50
.256008
3.9061250
.264834
3.7759519
1.03447
.966675
10
15
.258819
3.8637033
•267949
3.7320508
1.03528
.965926
75
10
.261628
3.8222251
.271069
3.6890927
1.03609
.965169
50
20
.264434
3.7816596
.274195
3.6470467
I.O369I
.964404
40
30
.267238
3^74i9775
•277325
3.6058835
1.03774
.963630
30
40
.^70040
3.7031506
.280460
3.5655749
1.03858
.962849
20
50
.272840
3.6651518
.283600
3.5260938
i.o39'44
.962059
10
16
•275637
3.6279553
.286745
3.4874144
1.04030
.961262
74
10
.278432
35915363
.289896
3.4495120
1.04117
.960456
50
20
.281225
35558710
.293052
3.4123626
1.04206
.959642
40
, 30
.284015
3.5209365
.296214
3.3759434
1.04295
.958820
30
40
.286803
3.48671 10
.299380
3.3402326
1.04385
•957990
20
50
.289589
3^4531735
•302553
3.3052091
1.04477
.957151
10
17
.292372
3.4203036
.305731
3.2708526
1.04569
•956305
73
10
•295152
3.3880820
.308914
3.2371438
1.04663
.955450
50
20
.297930
33564900
.312104
3.2040638
1.04757
•954588
40
30
.300706
3^3255095
.315299
3.1715948
1.04853
•953717
30
40
•303479
3.2951234
.318500
31397194
1.04950
•952838
20
50
.306249
3.2653149
.321707
3.I0842IO
1.05047
•951951
10
18
.309017
3.2360680
.324920
30776835
1.05146
•951057
72
10
.311782
3.2073673
.328139
3.0474915
1.05246
•950154
50
20
•314545
3.1791978
•331364
3.OI783OI
105347
.949243
40
30
•317305
31515453
.334595
2.9886850
1.05449
.948324
30
40
.320062
3.1243959
.337833
2.9600422
1.05552
.947397
20
50
.322816
30977363
•341077
2.9318885
1.05657
.946462
10
19
•325568
3^o7i5535
.344328
2.9042109
1.05762
•945519
71
10
•328317
3^0458352
.347585
2.8769970
1.05869
.944568
50
20
•331063
3.0205693
.350848
2.8502349
1.05976
•943609
40
30
•333807
2.9957443
.354119
2.8239129
1.06085
.942641
30
40
•336547
2.9713490
.357396
2.7980198
1.06195
.941666
20
50
•339285
2.9473724
.360680
2.7725448
1.06306
.940684
10
Cosine.
Secant.
Cotangent.
Tangent
Cosecant.
i
Sine.
Angle.
io6
MATHEMATICAL TABLES.
Angle.
Sine.
Cosecant.
Tangent.
Cotangent.
Secant.
Cosine.
20** O'
.342020
2.9238044
•363970
2.7474774
1. 06418
•939693
70° 0'
lO
•344752
2.9006346
.367268
2.7228076
I.0653I
.938694
50
20
.347481
2.8778532
•370573
2.6985254
1.06645
.937687
40
30
.350207
2.8554510
•373885
2.6746215
1. 06761
.936672
30
40
•352931
2.8334185
•377204
2.6510867
1.06878
.935650
20
50
•355651
2.8117471
.380530
2.627912I
1.06995
•934619
10
21
.358368
2.7904281
.383864
2.6050891
I.07115
•933580
69
10
.361082
2.7694532
.387205
2.5826094
^•07235
•932534
50
20
•363793
2.7488144
.390554
2.5604649
1.07356
.931480
40
30
.366501
2.7285038
.393911
2.5386479
1.07479
.930418
30
40
.369206
2.7085139
.397275
2.5171507
1.07602
.929348
20
50
.371908
2.6888374
.400647
2.4959661
1.07727
.928270
10
22
.374607
2.6694672
.404026
2.4750869
107853
.927184
68
10
■377302
2.6503962
.407414
2.4545061
1. 07981
.926090
50
20
•379994
2.6316180
.410810
2.4342172
1. 08109
.924989
40
30
.382683
2.6131259
.414214
2.4142136
1.08239
.923880
30
40
•385369
2.5949137
.417626
2.3944889
1.08370
.922762
20
SO
.388052
2.5769753
.421046
2.3750372
1.08503
.921638
10
23
.390731
2.5593047
.424475
2.3558524
1.08636
•920505
67
10
•393407
2.5418961
.427912
2.3369287
I.08771
.919364
50
20
.396080
2.5247440
•431358
2.3182606
1.08907
.918216
40
30
•398749
2.5078428
.434812
2.2998425
1.09044
.917060
30
40
.401415
2.491 1874
.438276
2.2816693
I.O9183
.915896
20
50
.404078
2.4747726
.441748
2.2637357
1.09323
.914725
10
24
.406737
2.4585933
•445229
2.2460368
1.09464
•913545
66
10
.409392
2.4426448
.448719
2.2285676
1.09606
•912358
50
20
.412045
2.4269222
.452218
2.2II3234
1.09750
.911164
40
30
.414693
2.4II42IO
.455726
2.1942997
1.09895
.909961
30
40
.417338
2.3961367
.459244
2.1774920
I.IOO4I
.908751
20
SO
.419980
2.3810650
.462771
2.1608958
I.IO189
.907533
10
25
.422618
2.3662016
.466308
2.1445069
I.IO338
.906308
65
10
.425253
2.3515424
•469854
2.I283213
1. 10488
•905075
50
20
.427884
2.3370833
•473410
2. 1 1 23348
1. 10640
•903834
40
30
•4305 1 1
2.3228205
•476976
2.0965436
I.IO793
•902585
30
40
•433^35
2.3087501
.480551
2.0809438
1. 10947
.901329
20
so
•435755
2.2948685
.484137
2.0655318
I.III03
.900065
10
26
•438371
2.281172O
.487733
2.0503038
I.II260
.898794
64
10
.440984
2.2676571
.491339
2.0352565
I.II419
.897515
50
20
•443593
2.2543204
.494955
2.0203862
I.II579
.896229
40
30
.446198
2.24II585
.498582
2.0056897
1. 1 1 740
.894934
30
40
.448799
2.2281681
.502219
I.99II637
I.I 1903
.893633
20
SO
•451397
2.2153460
.505867
1.9768050
1. 12067
.892323
10
•
Cosine.
Secant.
Cotangent.
Tangent.
Cosecant
Sine.
Angle.
SINES, COSINES, &C OF ANGLES.
107
Aflgk.
Sine.
Cosecant.
Tangent.
Cotangent.
Secant.
Cosine.
27*^0'
•453990
2.2026893
•509525
1. 9626105
1. 12233 ' ,891007
63° 0'
10
.456580
2.I901947
•513195
1.9485772
1. 1 2400
.889682
50
20
.459166
2.1778595
.516876
1.9347020
I.I2568
.888350
40
30
.461749
2.1656806
.520567
1. 9209821
1. 12738
.887011
30
40
.464327
2.1536553
.524270
I.9074147
1. 12910
.885664
20
50
.466901
2.I417808
.527984
1.893997 1
1. 13083
.884309
10
28
.469472
2.1300545
.531709
1.8807265
I.I3257
.882948
62
10
.472038
2.II84737
•535547
1,8676003
I 13433
.881578
50
20
.474600
2.1070359
•539195
I.8546159
1. 13610
.880201
40
30
.477159
2.0957385
.542956
I.8417409
I.I3789
.878817
30
40
•479713
2.0845792
.546728
1.8290628
1. 13970
.877425
20
50
.482263
2.0735556
•550515
1. 8164892
I.I4152
.876026
10
2g
.4848x0
2.0626653
.554309
1.8040478
114335
.874620
61
10
.487352
2.0519061
.558118
1. 7917362
1.14521
.873206
50
20
.4S989O
2.0412757
•561939
1.7795524
1.14707
.871784
40
30
.492424
2.0307720
.565773
1.7674940
1. 14896
.870356
30
40
•494953
2.0203929
.569619
1.7555590
1.15085
.868920
20
50
•497479
2.OIOI362
.573478
'.7437453
1.15277
.867476
10
30
.500000
2.0000000
.577350
1.7320508
i.15470
.866025
60
10
•502517
T.9899822
.581235
1.7204736
1.T5665
.864567
50
20
.505030
T. 98008 10
•585134
I.709OT16
1.15861
.863102
40
30
•507538
1.9702944
.589045
I.697663I
1. 16059
.861629
30
40
.510043
1.9606206
.592970
1.6864261
1.16259
.860149
20
50
•512543
1.95^0577
.596908
1.6752988
1.16460
.858662
10
31
.515038
1. 9416040
.600861
1.6642795
1.16663
.857167
59
10
•517529
1.9322578
.604827
1.6533663
1.16868
.855665
50
20
.520016
I.923OI73
.608807
1.6425576
1.17075
.854156
40
30
.522499
1.9138809*
.612801
1.6318517
1.17283
.852640
30
40
•524977
1.9048469
.616809
1.6212469
1.17493
.851117
20
50
.527450
I.8959138
.620832
1. 6107417
1.17704
.849586
10
32
.529919
1.8870799
.624869
1.6003345
1.17918
.848048
58
10
.532384
1.8783438
.628921
1.5900238
1.18133
.846503
50
20
.534844
1.8697040
.632988
1.5798079
1.18350
.844951
40
30
•537300
i.8611590
.637079
1.5696856
1.18569
.843391
30
40
•539751
1.8527073
.641167
15596552
1.18790
.841.825
20
50
.542197
1.8443476
.645280
'•5497155
1. 19012
.840251
. 10
33
.544639
1.8360785
.649408
1.5398650
1.19236
.838671
57
10
.547076
1.8278985
.653531
1. 5301025
1. 19463
837083
50
20
.549509
I.8198065
.657710
1. 5204261
1. 19691
.835488
. 40
30
•551937
I.81180IO
.661886
I.5108352
1.19920
.833886
30
40
.554360
1.8038809
.666077
I.5013282
1.20152
.832277
20
50
.556779
1.7960449
.670285
I.4919039
1.20386
.830661
10
OxBoe.
Secant.
Cotangent.
Tangent
Cosecant.
1
Sine.
Angle.
io8
MATHEMATICAL TABLES.
Angle.
Sine.
Cosecant
Tangent.
Cotangent.
Secant.
Co^e.
•
34° o'
•559193
1.7882916
•674509
1. 4825610
1.20622
.829038
56° 0'
lO
.561602
1.7806201
•678749
1.4732983
1.20859
.827407
50
20
.564007
1.7730290
.683007
I.464II47
1. 21099
.825770
40
30
.566406
I.7655173
.687281
1.4550090
1.21341
.824126
30
40
.568801
1.7580837
.691573
I.445980I
I.21584
.822475
20
50
•571191
1.7507273
.695881
1.4370268
1. 21830
.820817
10
35
•573576
1.7434468
.700208
1. 4281480
1.22077
.819152
55
10
•575957
1. 7362413
.704552
I.4193427
1.22327
.817480
50
20
•578332
1. 7291096
.708913
1. 4106098
1.22579
.815801
40
30
.580703
1.7220508
.713293
I.4OI9483
1.22833
.814116
30
40
.583069
I.7150639
.717691
13933571
1.23089
.812423
20
50
•5S5429
1. 7081478
.722108
1.3848355
1.23347
.810723
10
36
•587785
I.7013016
.726543
1.3763810
1.23607
.809017
54
10
.590136
1.6945244
.730996
^•3679959
1.23869
.807304
50
20
.592482
I.687815I
.735469
1.3596764
1. 24134
.805584
40
30
.594823
1. 681 1730
.739961
1.3514224
1.24400
.803857
30
40
•597159
1.6745970
.744472
1.3432331
1.24669
.802123
20
50
.599489
1.6680864
.749003
1.3351075
1.24940
.800383
10
37
.601815
I.661640I
.753554
1.3270448
1. 25214
.798636
53
10
.604136
16552575
.758125
1.3190441
1.25489
.796882
50
20
.606451
1.6489376
.762716
1.3 1 1 1046
1.25767
.795121
40
30
.608761
1.6426796
.767627
1.3032254
1.26047
•793353
30
40
.611067
1.6364828
.771959
1.2954057
1.26330
.791579
20
50
.613367
1.6303462
.776612
1.2876447
1. 26615
.789798
10
38
.615661
1.6242692
.781286
1.2799416
1.26902
.788011
52
10
.617951
I.618251O
.785981
1.2722957
I.2719I
.786217
50
20
.620235
1. 6122908
.790698
1.2647062
1.27483
.784416
40
30
.622515
1.6063879
.795436
1.2571723
1.27778
.782608
30
40
.624789
1. 6005416
.800196
1.2496933
1.28075
.780794
20
50
.627057
I.594751I
.804080
1.2422685
1.28374
.778973
10
39
.629320
I.589OI57
.809784
1.2348972
1.28676
.777146
51
10
•631578
1.5833318
.814612
1.2275786
1.28980
.775312
50
20
.633831
15777077
.819463
1.2203121
1.29287
.773472
40
30
.636078
1.5721337
.824336
1. 2130970
1.29597
.771625
30
40
.638320
I.566612I
.829234
1.2059327
1.29909
•769771
20
SO
.640557
I.5611424
.834155
1.1988184
1.30223
.767911
10
40
.642788
1.5557238
.839100
1.1917536
I.3054I
.766044
50
10
.645013
1.5503558
.844069
1. 1847376
1. 30861
.764171
50
20
.647233
1.5450378
.849062
1.1777698
1.3^83
.762292
40
30
.649448
1.5397690
.854081
1. 1 708496
I.31509
.760406
30
40
•651657
1.5345491
.859124
1. 1639763
1.31837
.758514
20
50
.653861
1.5293773
.864193
1.1571495
I.32168
.756615
10
Cosine.
Secant.
Cotangent.
Tangent.
Cosecant.
Sine.
Angle.
SINES, COSINES, &c. OF ANGLES.
109
Angle.
1
Sixie.
Cosecant.
Tangent.
Cotangent.
Secant.
Cosine.
«
41° 0'
.656059
I.524253I
.869287
1. 1503684
1.32501
•754710
49° 0'
10
.658252
I.519I759
.874407
1. 1436326
1.32838
.752798
50
20
.660439
I.514I452
.879553
I.I369414
I.33177
.750880
40
30
.662620
I.5091605
.884725
1. 1302944
I.33519
.748956
30
40
.664796
I.50422II
.889924
1. 1236909
1.33864
.747025
20
50
.666966
1.4993267
.895151
I.II71305
1.34212
.745088
10
142
.669131
1.4944765
.900404
I.II06125
1.34563
.743145
48
ID
.671289
1.4896703
.905685
I.IO41365
I.34917
.741195
50
20
.673443
1.4849073
.910994
1.0977020
1.35274
.739239
40
30
.675590
1. 4801872
.916331
I.O913085
1.35634
.737277
30
40
.677732
1.4755095
.921697
1.0849554
135997
.735309
20
50
.679868
1.4708736
.927091
1.0786423
1.36363
.733335
10
43 ex
.681998
1.4662792
.932515
1.0723687
1.36733
.731354
47
10
.684123
I.4617257
.937968
I.066134I
137105
.729367
50
20
.686242
I.4572127
.943451
I.0599381
1.37481
.727374
40
30
.688355
1.4527397
.948965
I.O53780I
1.37860
.725374
30
40
.690462
1.4483063
.954508
1.0476598
1.38242
.723369
20
50
.692563
1.4439 1 20
.960083
I.O415767
1.38628
.721357
10
44
.694658
1.4395565
.965689
1.0355303
1.39016
.719340
46
10
.696748
1.4352393
.971326
1.0295203
1.39409
.717316
50
20
.698832
1.4309602
.976996
I.O235461
1.39804
.715286
40
30
. 700909
1. 4267182
.982697
i.oi 76074
1.40203
.713251
30
40
.702981
I.4225134
.988432
I.OII7088
1.40606
.711209
20
i 50
1
.705047
1.4183454
.994199
1.0058348
1.41012
.709161
10
45
.707107
I.4142136
1. 000000
1. 0000000
1.41421
.707107
45
Cosine.
SecanL
Cotangent.
Tangent
Cosecant.
Sine.
Angle.
'
no
MATHEMATICAL TABLES.
TABLE No. IX.— LOGARITHMIC SINES, COSINES, TANGENTS,
AND COTANGENTS OF ANGLES FROM o^ TO 90^
Advancing by ic/, or onesixth of a degree.
Angle.
Sine.
Tangent
Cotangent.
. Cosine.
0°
0.000000
0.000000
Infinite.
10.000000
9o«
lo'
7.463726
7.463727
".536273
9.999998
50'
20
7.764754
7.764761
12.235239
9.999993
40
30
7.940842
7.940858
12.059142
9.999983
30
40
8.065776
8.065806
11.934194
9.999971
20
50
8.162681
8.162727
11.837273
9.9999S4
10
I
8.241855
8.24192I
11.758079
9.999934
89
10
8.308794
8.308884
11.691116
9.999910
50
20
8.366777
8.366895
11.633105
9.999882
40
30
8.417919
8.418068
11.58x932
9.999851
30
40
8.463665
8.463849
11.536151
9.999816
20
50
8.505045
8.505267
11.494733
9.999778
10
2
8.542819
8.543084
11.456916
9.999735
88
10
8.577566
8.577877
11.422123
9.999689
50
20
8.609734
8.610094
11.389906
9.999640
40
30
8.639680
8.640093
11.359907
9.999586
30
40
8.667689
8.668160
11.331840
9.999529
20
50
8.693998
8.694529
11.305471
9.999469
10
3
8.718800
8.719396
11.280604
9.999404
87
10
8.742259
8.742922
11.257078
9.999336
50
20
8. 7645 II
8.765246
11.234754
9.999265
40
30
8.785675
8.786486
11.213514
9.999189
30
40
8.805852
8.806742
11.193258
9.999110
20
SO
8.825130
8.826103
11.173897
9.999027
10
4
8.843585
8.844644
11.155356
9.998941
86
10
8.861283
8.862433
II. 137567
9.998851
50
20
8.878285
8.879529
1 1. 1 2047 1
9.998757
40
30
8.894643
8.895984
11.104016
9.998659
30
40
8.910404
8.9I1846
11.088154
9.998558
20
SO
8.925609
8.927156
11.072844
9.998453
10
5
8.940296
8.941952
11.058048
9.998344
85
10
8.954499
8.956267
11.043733
9.998232
50
20
8.968249
8.970133
11.029867
9.9981 16
40
30
8.981573
8.983577
II. 016423
9.997996
30
40
8.994497
8.996624
11.003376
9.997872
20
so
9.007044
9.009298
10.990702
9.997745
10
Cosine.
Cotangent.
Tangent
Sine.
Angle.
LOGARITHMIC SINES, TANGENTS, &C.
Ill
Aogie.
1
1 Sine.
1
Tangent
Cotangent.
Cosine.
6^
9.019235
9.021620
10.978380
9.997614
84°
lo'
9,031089
9.O33JS09
10.966391
9.997480
50'
20
9.042625
9.045284
10.954716
9.997341
40
30
9053859
9.056659
10.943341
9.997199
30
40
9.064806
9.067752
10.932248
9.997053
20
50
9.075480
9.078576
10.921424
9.996904
10
7
9.085894
9.089144
10.910856
9.996751
83
10
9.096062
9.099468
10.900532
9.996594
50
20
9.105992
9.109559
10.890441
9.996433
40
30
9. 1 15698
9.II9429
10.880571
9.996269
30
40
1 9125187
9.129087
10.870913
9.996100
20
50
9.134470
9138542
10.861458
9.995928
10
8
9I43SS5
9.147803
10.852197
9.995753
82
10
9152451
9.156877
10.843123
9.995573
50
20
9.161164
9165774
10.834226
9.995390
40
30
9.169702
9.174499
10.825501
9.995203
30
40
9.178072
9183059
IO.81694I
9.995013
20
50
9.186280
9.I91462
10.808538
9.994818
10
9
9194332
9199713
10.800287
9.994620
81
10
9.202234
9.207817
10.792183
9.994418
50
20
9.209992
9.215780
10.784220
9.994212
40
30
9.217609
9.223607
10.776393
9.994003
30
40
9.225092
9.231302
10.768698
9.993789
20.
50
9.232444
9.238872
IO.761128
9993572
10
10
9.239670
9.246319
10.753681
9.993351
80
10
9.246775
9.253648
10.746352
9.993127
50
20
' 9253761
9.260863
10.739137
9.992898
40
3<^
9.260633
9.267967
10.732033
9.992666
30
40
92673,95
9.274964
10.725036
9.992430
20
50
9.274049
9.281858
IO.718142
9.992190
10
II
9.280599
9.288652
10.71 1348
9.991947
79
10
9.287048
9.295349
10.704651
9.991699
50
20
9293399
9301951
10.698049
9.991448
40
30
9299655
9.308463
10.691537
9.991 193
30
40
9305819
9.314885
10.6851 15
9.990934
20
50
931 1893
9.321222
10.678778
9.990671
10
12
9.317879
9.327475
10.672525
9.990404
78
10
9.323780
9333646
10.666354
9.990134
50
20
9329599
9.339739
10.660261
9.989860
40
30
9335337
9.34575s
10.654245
9.989582
30
40
9340996
9.351697
10.648303
9.989300
20
50
9346779
9.357566
10.642434
9.989014
10
Cosine.
Cotangent.
Tangent
Sine.
Angle.
112
MATHEMATICAL TABLES.
Angle.
Sine.
Tangent.
Cotangent.
Cosine.
1
13°
9.352088
9.363364
10.636636
9.988724
If
10'
9357524
9369094
10.630906
9.988430
so'
20
9.362889
9.374756
10.625244
9988133
40
30
9.368185
9380354
10.619646
9.987832
30
40
9373414
9.385888
IO.614II2
9.987526
20
50
9378577
9.391360
10.608640
9.987217
10
14
9383675
9.396771
10.603229
9.986904
76
10
9.388711
9.402124
10.597876
9.986587
50
20
9393685
9.407419
10.592581
9.986266
40
30
9.398600
9.412658
10.587342
9.985942
30
40
9403455
9.417842
10.582158
9.985613
20
50
9.408254
9.422974
10.577026
9.985280
10
15
9.412996
9.428052
10.571948
9984944
75
10
9.417684
9.433080
10.566920
9.984603
50
20
9.422318
9438059
IO.56194I
9984259
40
30
9.426899
9.442988
10.557012
9.9839 1 1
30
40
9.431429
9.447870
10.552130
9983558
20
50
9435908
9.452706
10.547294
9.983202
10
16
9440338
9457496
10.542504
9.982842
74
10
9.444720
9.462242
10.537758
9.982477
50
20
9.449054
9.466945
10.533055
9.982109
40
30
9453342
9.471605
10.528395
9981737
30
40
9457584
9.476223
10.523777
9.981361
20
50
9.461782
9.480801
IO.519199
9.980981
10
17
9465935
9.485339
1 0.5 1 466 1
9.980596
73
10
9.470046
9489838
IO.51O162
9.980208
50
20
94741 15
9494299
10.505701
9.979816
40
30
9.478142
9.498722
10.501278
9979420
30
40
9.482128
9.503109
10.496891
9.979019
20
50
9.48607s
9.507460
10.492540
9.978615
10
18
9.489982
9.5II776
10.488224
9.978206
72
10
9.493851
9516057
10.483943
9977794
50
20
9.497682
9.520305
10.479695
9977377
40
30
9.501476
9.524520
10.475480
9976957
30
40
9.505234
9.528702
10.471298
9976532
20
50
9508956
9.532853
10.467147
9.976103
10
19
9.512642
9.536972
10.463028
9.975670
71
10
9.516294
9.541061
10.458939
9.975233
50
20
9.519911
9.545II9
10.454881
9.974792
40
30
9.523495
9.549149
10.450851
9.974347
30
40
9.527046
9.553149
10.446851
9.973897
20
50
9.530565
9.55712I
10.442879
9973444
10
Cosine.
Cotangent
Tangent.
Sine.
Angle.
LOGARITHMIC SINES, TANGENTS, &C
"3
Ai«le.
Sine.
Tangent.
Cotangent.
Cosme.
20^
9534052
9.561066
10.438934
9.972986
700
10'
9537507
9.564983
10.435017
9.972524
50'
20
9540931
9.568873
IO.43II27
9.972058
40
30
9544325
9.572738
10.427262
9.971588
30
40
9.547689
9.576576
10.423424
9.971113
20
50
9551024
9.580389
IO.41961I
9.970635
10
21
9554329
9.584177
10.415823
9.970152
69
10
9.557606
9.587941
10.412059
9.969665
50
20
9.56085s
9.591681
10.408319
9.969173
40
30
9.564075
9.595398
10.404602
9.968678
30
40
9.567269
9.599091
10.400909
9.968178
20
50
9.570435
9.602761
10.397239
9.967674
10
22
9573575
9.606410
10.393590
9.967166
68
10
9.576689
9.610036
10.389964
9.966653
50
20
9.579777
9.61364I
10.386359
9.966136
40
30
9.582840
9.617224
10.382776
9965615
30
40
9.585877
9.620787
10.379213
9.965090
20
50
9.588890
9.624330
10.375670
9.964560
10
23
9.591878
9.627852
10.372148
9.964026
67
10
9.594842
9.631355
10.368645
9.963488
50
20
9.597783
9.634838
10.365162
9.962945
40
30
9.600700
9.638302
10.361698
9.962398
30
40
9.603594
9.641747
10.358253
9.961846
20
50
9.606465
9.645174
10.354826
9.961290
10
M
9.609313
9.648583
IO.351417
9.960730
66
10
9.6 1 2 140
9.651974
10.348026
9.960165
SO
20
9.614944
9.655348
10.344652
9959596
40
30
9.617727
9.658704
10.341296
9959023
30
40
9.620488
9.662043
10.337957
9.958445
20
50
9.623229
9.665366
10.334634
9.957863
10
25
9.625948
9.668673
10.331328
9.957276
65
10
9.628647
9.671963
10.328037
9.956684
50
20
9.631326
9.675237
10.324763
9.956089
40
30
9.633984
9.678496
10.321504
9955488
30
40
9.636623
9.681740
10.318260
9954883
20
50
9.639242
9.684968
10.315032
9954274
10
26
9.641842
9.688182
IO.3I1818
9953660
64
10
9.644423
9691381
10.308619
9953042
SO
20
9.646984
9.694566
10.305434
9.952419
40
30
9.649527
9.697736
10.302264
9951791
30
40
9.652052
9.700893
10.299107
995"S9
20
50
9.654558
9.704036
10.295964
9.950522
10
Cosine.
Cotangent.
Tangent
Sine.
Angle.
8
"4
MATHEMATICAL TABLES.
Angle.
Sine.
Tangent.
Cotangent.
Cosine.
27°
9.657047
9.707166
10.292834
9.949881
63^
10'
9659517
9.710282
10.289718
9949235
50'
20
9.661970
9.713386
10.286614
9.948584
40
30
9.664406
9.716477
10.283523*
9.947929
30
40
9.666824
9719555
10.280445
9.947269
20
50
9.669225
9.722621
10.277379
9.946604
10
28
9.671609
9.725674
10.274326
9945935
62
10
9673977
9.728716
10.271284
9.945261
50
20
9.676328
9731746
10.268254
9.944582
40
30
9.678663
9734764
10.265236
9943899
30
40
9.680982
9737771
10.262229
9.943210
20
50
9.683284
9.740767
10.259233
9942517
10
29
9685571
9743752
10.256248
9.941819
61
10
9.687843
9.746726
10.253274
9.941117
50
20
9.690098
9.749689
IO.2503II
9.940409
40
30
9.692339
9.752642
10.247358
9.939697
30
40
9.694564
9755585
10.244415
9.938980
20
50
9.696775
9.758517
10.241483
9938258
10
30
9.698970
9.761439
10.238561
9937531'
60
10
9.70II51
9.764352
10.235648
9.936799
50
20
9703317
9767255
10.232745
.9.936062
40
30
9.705469
9.770148
10.229852
9935320
30
40
9.707606
9.773033
10.226967
9934574
20
SO
9.709730
9775908
10.224092
9.933822
10
31
9.711839
9778774
10.221226
9.933066
59
10
9713935
9.781631
10.218369
9.932304
50
20
9.716017
9784479
IO.21552I
9931537
40
30
9718085
9787319
IO.212681
9.930766
30
40
9.720140
9.790151
10.209849
9.929989
20
50
9.722181
9792974
10.207026
9.929207
10
32
9.724210
9.795789
10.2042 1 1
9.928420
58
10
9.726225
9.798596
10.201404
9.927629
50
20
9.728227
9.801396
10.198604
9.926831
40
30
9.730217
9.804187
IO.I95813
9.926029
30
40
9732193
9.806971
10.193029
9.925222
20
50
9734157
9.809748
10.190252
9.924409
10
33
9.736109
9.812517
10.187483
9923591
57
10
9.738048
9.815280
10.184720
9.922768
50
20
9739975
9.818035
IO.181965
9.921940
40
30
9.741889
9.820783
IO.I79217
9.921107
30
40
9743792
9823524
10.176476
9.92026^
20
50
9.745683
9.826259
IO.I7374I
9.919424
10
Cosine.
Cotangent.
Tangent.
Sine.
Angle.
LOGARITHMIC SINES, TANGENTS, &C
115
Angle.
Sine.
Tangent
Cotangent
Cosine.
' 34
9.747562
9.828987
IO.1710I3
9.918574
56° ,
10'
9.749429
9.831709
. IO.16829I
9.917719
50'
20
9.751284
9834425
10.165575
9.916859
40
30
9753128
9837134
10.162866
9915994
30
40
9.754960
9.839838
IO.160162
9915123
20
50
9.756782
9842535
10.157465
9.914246
10
35
9758591
9.845227
10.154773
9.913365
55
10
9.760390
9.847913
10.152087
9.912477
50
20
9.762177
9850593
10.149407
9.91 1584
40
30
9763954
9.853268
10.146732
9.910686
30
40
9.765720
9855938
10.144062
9.909782
20
50
9767475
9.858602
IO.I41398
9.908873
10
36
9.769219
9.861261
10.138739
9.907958
54
10
9.770952
9863915
10.136085
9.907037
50
20
9.77267s
9.866564
10.133436
9.9061 11
40
30
9774388
9.869209
IO.I3079X
9.905179
30
40
9.776090
9.871849
IO.I28151
9.904241
20
50
1
9777781
9.874484
IO.I25516
9.903298
10
1
37
9779463
9.877114
10.122886
9.902349
53
10
1 9781134
9.879741
10.120259
9.901394
50
20
1 9782796
9.882363
IO.II7637
9.900433
40
30
9.784447
9.884980
10. 1 1 5020
9.899467
30
40
9.786089
9.887594
10. 1 12406
9898494
20
50
9.787720
9.890204
10.109796
9.897516
10
38
9.789342
9.892810
IO.IO719O .
9.896532
52
10
9790954
9.895412
10.104588
9.895542
50
20
9792557
9.898010
IO.IOI99O
9.894546
40
30
9794150
9.900605
10.099395
9.893344
30
40
9795733
9.903197
10.096803
9.892536
20
50
9.797307
9905785
10.094215
9.891523
10
39
9798872
9.908369
IO.O9163I
9.890503
51
10
9.800427
9.910951
10.089049
9.889477
50
20
9.801973
9913529
10.086471
9.888444
40
1 3<^
9.803511
9.916104
10.083896
9.887406
30
1 ^^
40
9.805039
9.918677
10.081323
9.886362
20
50
9806557
9.921247
10.078753
9.885311
10
40
9.808067
9.923814
10.076186
9.884254
50
10
9.809569
9.926378
10.073622
9.883191
50
20
9.811061
9.928940
10.071060
9.88212I
40
30
9.812544
9931499
10.068501
9.881046
30
40
9.814019
9934056
10.065944
9.879963
20
50
9815485
9.93661 1
10.063389
9.878875
10
Coune.
Cotangent.
T.ingcnt.
Sine.
Angle.
.:j,«rrf'^f"
ri6
MATHEMATICAL TABLES.
Angle.
Sine.
Tangent
Cotangent.
Cosine.
1
41°
9.816943
9939163
10.060837
9.877780
49°
10'
9.818392
9941713
10.058287
9.876678
50'
20
9.819832
9.944262
10.055738
9875571
40
30
9.821265
9.946808
10.053192
9.874456
30
40
9.822688
9949353
10.050647
9.873335
20
50
9.824104
9.951896
10.048104
9.872208
10
42
9825511
9954437
10.045563
9.871073
48
10
9.826910
9956977
10.043023
9.869933
50
20
9.828301
9959516
10.040484
9.868785
40
30
9.829683
9.962052
10.037948
9.867631
30
40
9.831058
9.964588
10.035412
9.866470
20
50
9832425
9.967123
10.032877
9.865302
10
43
9833783
9.969656
10.030344
9.864127
47
10
9835134
9,972188
10.027812
9.862946
50
20
9.836477
9.974720
10.025280
9861758
40
30
9.837812
9.977250
10.022750
9.860562
30
40
9.839140
9.979780
10.020220
9.859360
20
50
9.840459
9.982309
10.01 7691
9.85815I
10
44
9.841771
9984837
IO.OI5163
9856934
46
10
9.843076
9987365
10.012635
9.855711
SO
20
9.844372
9.989893
IO.OIOIO7
9.854480
40
30
9.845662
9.992420
10.007580
9853242
30
40
9.846944
9.994947
10.005053
9851997
20
SO
9.848218
9997473
10.002527
9850745
10
45
9.849485
10.000000
10.000000
9.849485
45
Cosine.
Cotangent.
Tangent.
Sine.
Angle.
RHUMBS, OR POINTS OF THE COMPASS.
117
TABLE No. X.— RHUMBS, OR POINTS OF THE COMPASS.
Points.
H
>i
Va
I
2
3
sH
4
4H
6^
8
Angles.
2^48' 45'
5 37 30
8 26 15
II 15 o
U 3 45
16 52 30
19 41 15
22 30 o
25 18 45
28 7 30
30 56 15
33 45 o
36 33 45
39 22 30
42 n 15
45 o o
47 48 45
50 37 30
53 26 15
56 15 o
59 3 45
61 52 30
64 41 15
67 30 o
70 18 45
73 7 30
75 56 15
78 45 o
81 ZZ 45
84 22 30
87 II 15
90 o o
NORTH.
N ^ E
N 5^
 /* E
N ^ E
N by E
Nby E ;^
N by E J^ E
N by E ^ E
NNE
NNE ^ E
NNE ^ E
NNE ^ E
NE by N
NE^N
'' N
N
NE>52
NE
NE J^ E
NE J^ E
NE J^ E
NE by E
ENE ^ N
ENE y2 N
ENE ]/{ N
ENE
ENE ^ E
ENE ^ E
ENE y^ E
Eby N
E% N
E li N
E^N
EAST.
NORTH.
SOUTH.
N ^ W
N ^ W
N J^ W
N by W
N by w ^ w
N by w J^ w
N by w ^ w
NNW
NNW j5^ W
NNW j5 W
NNW y^ W
NW by N
NW ^ N
NW i^ N
NW^ N
NW.
NW l^ W
NW J^ W
NW J^ W
NW by w
WNW y^ N
WNW ^ N
WNW J^ N
WNW
WNW ^ W
WNW J^ W
WNW y^ W
. w by N
W^ N
w J4 N
W 5j( N
WEST.
S )4 E
S >^ E
S^E
s by E
E
sby E
s by E ^ E
s by £ )^ E
SSE
SSE ^ E
SSE Yi E
SSE J^ E
SE by s
SE ^ S
SE ^ S
SE ^ S
SE
SE l^ E
SE ^ E
SE ^ E
SE by E
ESE ^ S
ESE y2 S
ESE^ S
ESE
ESE % E
ESE J4 E
ESE y^ E
E by S
E ^ S
E^ S
E j^ S
EAST.
SOUTH.
S
s
s^w
s>^ w
s^w
s by w
sby w ^ w
by w J^ w
by w ^ w
ssw
ssw ^ w
ssw y^ w
ssw y^ w
sw by s
sw^ s
sw ^ s
sw i^ s
sw
sw^ w
sw J4 w
sw ^ w
swby w
wsw y^ s
wsw J4 s
wsw^ s
wsw
wsw 5^ w
wsw J^ w
wsw y^ w
w by s
w^ s
w J4s
w j^ s
WEST.
MATHEMATICAL TABLES.
TABLE No. XL— RECIPROCALS OF NUMBERS
RECIPROCALS OF NUMBERS.
119
No.
57
58
59
[60
61
62
63
64
66
67
68
69
70
71
72
73
74
75
76
77
73
79
80
81
82
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
200
201 I
Reciprocal.
006369
006329
006289
006250
0062 I I
006173
006135
006098
006061
006024
005988
005952
005917
005882
005848
005814
005780
005747
005714
005682
005650
005618
005587
005556
005225
005495
005464
005435
005405
005376
005348
005319
005291
005263
005236
005208
OO5181
005155
005128
005102
005076
005051
005025
005000
004975
No.
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232.
233
234
235
236
237
238
239
240
241
242
243
244
245
246
ReciprocaL
004950
004926
004902
004878
004854
00483 I
004808
004785
004762
004739
0047 I 7
004695
004673
00465 1
004630
004608
004587
004566
004545
004525
004505
004484
004464
004444
004425
004405
004386
004367
004348
004329
004310
004292
0042 74
004255
004237
004219
004202
004184
004167
004149
004132
004115
004098
004082
004065
No.
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
Reciprocal.
I
004049
004032
004016
004000
003984
003968
003953
003937
003922
003906
003891
003876
003861
003846
003831
003817
003802
003788
003774
003759
003745
003731
003717
003704
003690
003676
003663
003650
003636
003623
003610
003597
003584
003571
003559
003546
003534
003522
003509
003497
003484
003472
003460
003448
003436
No.
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
332
333
334
335
336
Reciprocal.
.003425
.003413
.003401
.003390
.003378
.003367
.003356
.003344
.003333
.003322
.003311
.003301
.003289
.003279
.003268
.003257
.003247
.003236
.003226
.003215
.003205
.003195
.003185
.003175
.003165
.003155
.003145
.003135
.003125
.003115
.003106
.003096
.003086
.003077
.003067
.003058
.003049
.003040
.003030
.003021
.003012
.003003
.002994
.002985
.002976
I
1 20
MATHEMATICAL TABLES.
No.
337
338
339
380
381
Reciprocal.
.002967
.002959
.002950
340
.002941
341
.002933
342
.002924
343
.002915
344
.002907
345
.002899
346
.002890
347
.002882
348
.002874
349
.002865
350
.002857
35J
.002849
352
.002841
353
.002833
354
.002825
355
.002817
356
.002809
357
.002801
358
.002793
359
.002786
360
.002778
361
.002770
362
.002762
363
.002755
364
.002747
365
.002740
366
.002732
367
.002725
368
.002717
369
.002710
370
.002703
371
.002695
372
.002688
373
.002681
374
.002674
375
.002667
376
.002660
377
.002653
378
.002646
379
.002639
.002632
.00^625
No.
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
Redprocal.
.002618
.002611
.002604
.002597
.002591
.002584
.002577
.002571
.002564
.002558
.002551
.002545
.002538
.002532
.002525
.002519
.002513
.002506
.002500
.002494
.002488
.002481
.002475
.002469
.002463
.002457
.002451
.002445
.002439
.002433
.002427
.002421
.002415
.002410
.002407
.002398
.002392
.002387
.002381
.002375
.002370
.002364
.002358
.002353
.002347
No.
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
Reciprocal.
.002342
.002336
.002331
,002326
.002320
.002315
.002309
.002304
.002299
.002294
.002288
.002283
.002278
.002273
.002268
.002262
.002257
.002252
.002247
.002242
.002237
.002232
.002227
.002222
.002217
.002212
.002208
.002203
.002198
.002193
.002188
.002183
.002179
.002174
.002169
.002165
.002160
.002155
.002151
.002146
.002141
,002137
.002132
.002128
.002123
No.
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
5x5
516
Reciprocal.
.002119
.002114
.002110
.002105
.002101
.002096
.002092
.002088
.002083
.002079
.002075
.002070
.002066
.002062
.002058
.002053
.002049
.002045
.002041
.002037
.002033
.002028
.002024
.002020
.002016
.002012
.002008
.002004
.002000
.001996
.001992
.001988
.001984
.001980
.001976
.001972
.001969
.001965
.001961
.001957
.001953
.001949
.001946
.001942
.001938
RECIPROCALS OF NUMBERS.
121
Ko.
518
520
521
522
526
528
529
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
RcaprocaL
.001934
.001931
.001927
.001923
.001919
.001916
.001912
.001908
.001905
.001901
.001898
.001894
.001890
.001887
.001883
.001880
.001876
.001873
.001869
.001866
.001862
.001859
.001855
.001852
.001848
.001845
.001842
.001838
.001835
.001832
.001828
.001825
.001821
.001818
.001815
.001812
.001808
.001805
.001802
.001799
.001795
.001792
.001789
.001786
.001783
No.
562
563
564
565
566
567
568
569
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
RedprocaL
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
,00
.00
.00
.00
.00
.00
.00
.00
.00
.00
79
76
73
70
67
64
61
57
54
51
48
45
42
39
36
33
30
27
24
21
18
15
12
09
06
04
01
698
695
692
689
686
684
681
678
675
672
669
667
664
661
658
656
653
650
No.
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
Reciprocal.
.001647
.001645
.001642
.001639
.001637
.001634
.001631
.001629
.001626
.001623
.001621
.001618
.001616
.001613
.001610
.001608
.001605
.001603
.001600
.001597
.001595
.001592
.001590
.001587
.001585
.001582
.001580
.001577
•001575
.001572
.001570
.001567
.001565
.001563
.001560
.001558
.001555
•001553
.001550
.001548
.001546
.001543
.001541
.001538
.001536
No.
Reciprocal.
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
.001534
.001531
.001529
.001527
.001524
.001522
.001520
.001517
.001515
.001513
.001511
.001508
.001506
.001504
.001502
.001499
.001497
.001495
.001493
.001490
.001488
.001486
.001484
.001481
.001479
.001477
.001475
.001473
.001471
.001468
.001466
.001464
.001462
.001460
.001458
.001456
.001453
.001451
.001449
.001447
.001445
.001443
.001441
.001439
.001437
122
MATHEMATICAL TABLES.
No.
Reciprocal.
697 .001435
698 .001433
699 .001431
rOO .001429
roi .001427
'02 .001425
'03 .001422
^04 .001420
ro5 .001418
'06 .001416
07 .001414
ro8 .001412
r09 .001410
10 .001408
11 .001406
12 .001404
13 .001403
14 .001401
15 .001399
16 .001397
17 .001395
18 .001393
19 .001391
'20 .001389
'21 .001387
22 .001385
23 .001383
24 .001381
25 .001379
26 .001377
27 .001376
28 .001374
29 .001372
^30 .001370
'31 .001368
'32 .001366
'33 .001364
'34 .001362
'35 .001361
36 .001359
r37 .001357
38 .001355
^39 .001353
r40 .001351 I
'41 .001350 !
No.
Reciprocal.
742
.001
743
.001
744
.001
745
.001
746
.001
747
.001
748
.001
749
.001
750
.001
751
.001
752
.001
753
.001
754
.001
755
.001
756
.001
757
.001
758
.001
759
.001
760
.001
761
.001
762
.001
763
.001
764
.001
765
.001
766
.001
767
.001
768
.001
769
.001
770
.001
771
.001
772
.001
773
.001
774
.001
775
.001
776
.001
777
.001
778
.001
779
.001
780
.001
781
.001
782
.001
783
.001
784
.001
785
.001
786
.001
348
346
344
342
340
339
337
335
333
332
328
326
325
323
321
319
318
316
314
312
311
309
307
305
304
302
300
299
297
295
294
292
290
289
287
285
284
282
280
279
277
276
274
272
No.
787
788
789
830
831
Reciprocal.
.001271
.001269
.001267
790
.001266
791
.001264
792
.001263
793
.001261
794
.001259
795
.001258
796
.001256
797
.001255
798
.001253
799
.001251
800
.001250
801
.001248
802
.001247
803
.001245
804
.001244
805
.001242
806
.001241
807
.001239
808
.001238
809
.001236
Bid
.001235
811
.001233
812
.001232
813
.001230
814
.001229
815
.001227
816
.001225
817
.001224
818
.001222
819
.OOI22I
820
.001220
821
.001218
822
.001217
823
.001215
824
.001214
825
.001212
826
.001211
827
.001209
828
.001208.
829
.001206
.001205
.001203
No.
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
Reciprocal.
.00:
.00]
.00:
.00:
.00:
.00:
.00:
.00:
.00:
.00:
.00:
.00:
.00;
.00:
.00;
.00;
.00;
.00;
.00]
.00:
.00]
.00:
.OOj
.00:
.00;
.00:
.00:
.00:
.00;
.00:
.OOJ
.OOJ
.00;
.00:
.00;
.00:
.00
.00:
.00:
.00;
.00:
.00:
.00
.00;
.00:
202
200
199
198
196
195
193
192
190
189
188
186
185
183
182
181
179
178
176
175
174
172
171
I 70
168
167
166
164
^^3
161
160
159
157
156
155
153
152
151
149
148
147
145
144
143
142
RECIPROCALS OF NUMBERS.
123
Now
Rec ip io ca L
877
.001
878
.001
879
.OOI
880
.001
S8I
.001
SS2
.001
883
.001
884
.001
885
.001
886
.001
887
.OOI
888
.001
889
.001
Sgo
.001
891
.001
892
.001
893
.001
894
.001
895
.001
896
.001
897
.001
898
.001
899
.001
QOO
.001
901
.001
902
.001
903
.001
904
.001
905
.001
906
.001
907
.001
140
139
138
136
135
134
133
131
130
129
127
126
125
124
122
121
120
119
118
116
"5
114
112
III
IIO
109
107
106
104
103
No.
908
909
910
911
912
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
931
932
933
934
935
936
937
938
Reciprocal.
.001101
.001100
.001099
.001098
.001096
.001095
.001094
.001093
.001092
.001091
.001089
.001088
.001087
.001086
.001085
.001083
.001082
.001081
.001080
.001079
.001078
.001076
.001075
.001074
.001073
.001072
.001071
.001070
.001068
.001067
.001066
No.
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958.
959
960
961
962
963
964
965
966
967
968
969
RcoprocaL
.001065
.001064
.001063
.001062
.001060
.001059
.001058
.001057
.001056
.001055
.001054
.001053
.001052
.001050
.001049
.001048
.001047
.001046
.001645
.001044
.001043
.001042
.001041
.001040
,001038
.001037
.001036
.001035
.001034
.001033
.001032
No.
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
Reciprocal.
,001031
.001030
.001029
.001028
.001027
.001026
.001025
.001024
.001022
.001021
.001020
.001019
.001018
.001017
.001016
.001015
.001014
.001013
.001012
.001011
.001010
.001009
.001008
.001007
.001006
.001005
.001004
.001003
.001002
.001001
.001000
WEIGHTS AND MEASURES.
WATER AND AIR AS STANDARDS FOR WEIGHT AND MEASURE.
WATER AS A STANDARD.
There are four notable temperatures for water, namely,
32° R, or 0° C. = the freezing point, under one atmosphere.
3 9°. I or 4° = the point of maximum density.
62° or 1 6°. 66 = the British standard temperature.
212° or 100° = the boiling point, under one atmosphere.
The temperature 62° F. is the temperature of water used in calculating
the specific gravity of bodies, with respect to the gravity or density of
water as a basis, or as unity. In France, the temperature of maximum
density, 39°.! F., or 4° C, is used for this purpose, for solids.
Weight of one cubic foot of Pure Water,
At 32° F. =■ 62.418 pounds.
At 39°. I = 62.425 „
At 62° (Standard temperature) = 62.355 »
At 212° = 59.640 „
The weight of a cubic foot of water is, it may be added, about 1000
ounces (exactly 998.8 ounces), at the temperature of maximum density.
The weight of water is usually taken in round numbers, for ordinary
calculations, at 62.4 lbs. per cubic foot, which is the weight at 52^.3 F. ; or
it is taken at 62)^ lbs. per cubic foot, where precision is not required, equal
to 1^^ lbs.
The weight of a cylindrical foot of water at 62° F. is 48.973 pounds.
Weight of one cubic inch of Pure Water,
At 32® F. = .03612 pounds, or 0.5779 ounce.
At39°.i =.036125 „ ,,0.5780 „
At 62° =.03608 „ ,,0.5773 „ or 252.595 grains.
At 212** = .03451 „ „ 0.5522 „
The weight of one cylindrical inch of pure water at 62** F. is 02833
pound, or 0.4533 ounce.
WATER AND AIR AS STANDARDS. 1 25
Volume of one pound of Pure Water,
At 32** F. = .016021 cubic foot, or 27.684 cubic inches.
At 39^1 = .016019 „ „ 27.680 „
At 62"* = .016037 „ „ 27.712 „
At 212** = .016770 „ „ 28.978 „
The volume of one ounce of pure water at 62** F. is 1.732 cubic inches.
The Gallon.
The weight of one gallon of water at the standard temperature, 62*^ F.,
is 10 pounds, and the correct volume is 0.160372 cubic foot, or 277.123
cubic inches. But in an Act of Parliament, now partly repealed, which
came into force in 1825, the volume of one gallon is stated to be 277.274
cubic inches; this is the commonly accepted volume.
The volume of 10 pounds of water at 62° F. is, therefore, to the volume
of the imperial gallon, as i to 1.000545.
And, the weight of an imperial gallon of water at 62° F. is 10.00545
pounds avoirdupois j or 10 pounds 38.15 grains.
One cubic foot of water contams 6.2355 gallons, or approximately 6^
gaDons.
The volume of water at 62*" F., in cubic inches, multiplied by .00036,
giv« the capacity in gallons.
The capacity of one gallon is equal to one square foot, two inches deep
nearly (exactly 1.924 inches); or to one circular foot, 2j^ inches deep
nearly (exactly 2.45 inches).
One ton of water at 62° F. contains 224 gallons.
Other Measures of Water.
Volume of given weights of water, at 62.4 pounds per cubic foot: —
I ton 3590 cubic feet.
I cwt 1.795 »
I quarter 449 „
, r .016 cubic foot, or
' P^^^ • J 27.692 cubic inches.
I ounce I73I »
I tonne, at 39°.! F 353I56 cubic feet.
1 1 ..or? f 0353 cubic foot, or
I kdogiamme, at 39°.i F  61.0^5 cubic inches.
I tonne, at 52;.3 F. ) 35.330 cubic feet
(62.4 pounds per cubic foot) J ^^ ^^
Thirtysix cubic feet, or i^ cubic yards, of water, at 62.4 pounds per
tabic foot, being at the temperature 52^.3 F., weigh about one ton (exactly
6.4 pounds more).
Ctae cubic yard, or twentyseven cubic feet, of water weighs about
15 cwt, or ^ ton (exactly 4.8 pounds more).
One cubic metre of water is equal in volume to 35.3156 cubic feet,
or 1.308 cubic yard, or 220.09 gallons; and, at 62.4 pounds per cubic foot,
it weighs i ton nearly (exactly 36.3 pounds less). It is nearly equivalent
126 WEIGHTS AND MEASURES.
to the old English tun of 4 hogsheads — 210 imperial gallons, and is a
better unit for measuring sewage or watersupply than the gallon.
The cubic metre is generally used on the Continent for such measurements.
A pipe one yard long holds about as many pounds of water as the square
of its diameter in inches (exactly 2 per cent. more).
Pressure of Water.
A pressure of one lb. per square inch is exerted by a column of water
2.3093 feet, or 27.71 inches high, at 62® F,; and a pressure of one atmos
phere, or 14.7 lbs. per square inch, is exerted by a column of water
33.947 feet high, or 10.347 metres, at 62° F.
A column of water at 62"^ F., one foot high, presses on the base with a
force of 0.433 lb., or 6.928 ounces per square inch. A column 100 feet
high presses with a force of 43^ lbs. per square inch. A column one
metre high presses with a force of 1.422 lbs. per square inch.
A column of water one inch high, presses on the base with a force of
0.5773 ounce per square inch, or 5.196 lbs. per square foot.
A column of water one mile deep, weighing 62.4 pounds per cubic foot,
presses on the base with a force of about one ton per square inch (fresh
water exacdy 48 lbs. more; sea water exactly 107.5 ^^s. more).
Water is hardly compressible under pressure. Experiment appears to
show that for each atmosphere of pressure it is condensed 47)^ millionths
of its bulk.
Seawater.
One cubic foot of average seawater, at 62** F., weighs 64 pounds, and
the weight of fresh water is to that of seawater as 39 to 40, or as i to 1.026.
Thirtyfive cubic feet of seawater weighs one ton.
One cubic yard of seawater weighs i^yi cwt nearly (8 lbs. less).
One cubic metre of seawater weighs fully one ton (20 lbs. more).
Average seawater is composed as follows : —
Per xoo paits. Per zoo parts.
Chloride of sodium • (common salt), 2.50
Sulphuret of magnesium, 0.53
Chloride of magnesium, 0.33
Carbonate of lime, )
Carbonate of magnesia, J
Sulphate of lime, o.oi
Solid matter, say, 3.40
Water, 96.60
100.00
»
I
showing that seawater contains ^^^th part of its weight of solid matter in
solution.
According to R^clus, the mean specific gravity of seawater is 1.028. In
the Mediterranean Sea, it is 1.029; in the Black Sea, 1.016. The mean
quantity of salts, or solid matter, in solution, is 3.44 percent., threefourths
of which is common salt In the Red Sea, the water contains 4.3 per cent. •
in the Baltic Sea, 5 per cent. ; and at Cronstadt, 2 per cent
WATER AND AIR AS STANDARDS. 12/
Ice and Snow,
One cubic foot of ice at 32° F. weighs 57.50 lbs.
One pound of ice at 32° F. has a volume of .0174 cubic foot, or 30.067
cubic inches.
The volume of water at 32° F. is to that of ice at 32** F., as i.ooo to
1.0855; ^^ expansion in passing into the sohd state being above 8)^ per
cent of the volume of water.
The specific density of ice is 0.922, that of water at 62° F. being = i.
The melting point of ice is ^2"^ F., or 0° C, under the ordinary atmos
pheric pressure, of 14.7 lbs. per square inch. Under greater pressure the
melting point is lower, being 'at the rate of .0133° F. for each additional
atmosphere of pressure.
The specific heat of ice is .504, that of water being = i.
One cubic foot of fresh snow weighs 5.20 lbs. Snow has 12 timeis the
bulk of water, and its specific gravity is .0833.
French and English Measures of Water,
One litre of water is equal to 0.2201 gallon, or 1.761 pints: about
i:^ pints. One gallon is equal to 4.544 litres, and one pint is .568 litre.
One litre of water at 39'^.! F., or 4° C, the temperature of maximum
density, weighs one kilogramme, or 2.2046 lbs.; at the temperature 62° F.,
or 1 6°. 7 C, it weighs 2.202 lbs.
looo litres = one cubic metre, equal to 35.3156 cubic feet; and, at
39^1 F., or 4** C, weigh 1000 kilogrammes, or one ton nearly (35.4 lbs. less).
AIR AS A STANDARD.
The mean pressure of the atmosphere at the level of the sea, is equal
to 14.7 lbs. per square inch, or* 21 16.4 lbs. per square foot; or to 1.0335
kilogrammes per square centimetre. This is called one atmosphere of
pressure. The following are measures of pressures (see also pages 1 45, 158): —
One atmosphere of pressure : — (i.) A column of air at 32° F., 27,801 feet,
or about 5j^ miles high, of uniform density, equal to that of air at the level
of the sea. (2.) A column of mercury at 32° F., 29.922 inches or 76 centi
metres high; nearly 30 inches. At 62° F., the height is 30 inches. (3.) A
column of water at 62** F., 33.947 feet or 10.347 metres high; nearly 34 feet.
A pressure of i lb. per square inch: — (i.) A column of air at 32° F.,
189 1 feet high, of uniform density as above. (2.) A column of mercury at
32' F., 2.035 inches or 51.7 millimetres high. At 62° F., the height is 2.04
inches. (3.) A column of water at 62° F., 2.31 feet or 27.72 inches high.
A pressure of i lb. per square foot: — (i.) A column of air at 32® F., 13.13
feet high, of uniform density as above. (2.) A column of mercury at 32° F.,
.0141 inch or .359 millimetre high. At 62° F., the height is .01417 inch.
(3.) A column of water at 62° F., .1925 inch high.
The density, or weight of one cubic foot of pure air, under a pressure
of one atmosphere, or 14.7 lbs. per square inch, is
At 32° F., = .080728 pound, or 1.29 ounce, or 565.1 grains.
At62**F., = .076097 „ „ 1.217 „ „ 53;2.7 „
The weight of a litre of pure air, under one atmosphere, at 32° F., is
1.293 gnunmesy or 19.955 grains.
128 WEIGHTS AND MEASURES.
The weight of air, compared with that of water at three notable tempera
tures, and at 5 2°. 3, under one atmosphere, is as follows: —
773.2 times the weight of air at 32° F.
77327 » » »f
772*4 » » »
819.4 „ „ 62^
820 „ „ „
The volume of one pound of air at 32** F., and under one atmosphere of
pressure, is 12.387 cubic feet. The volume at 62° F., is 13. 141 cubic feet.
The specific heat of air at constant pressure is .2377, and at constant
volume .1688, that of water being = i.
Weight
of water at
: 32° F.,
39". I,
62°,
62°,
»>
»
Sa^S.
GREAT BRITAIN AND IRELAND.— IMPERIAL WEIGHTS
AND MEASURES.
The origin of English measures is the grain of com. Thirtytwo grains
of wheat, dried and gathered from the middle of the ear, weighed what was
called one penn)rweight; 20 pennyweights were called one ounce, and
20 ounces one pound. Subsequently, the pennyweight was divided into
24 grains. Troy weight was afterwards introduced by William the Conqueror,
from Troyes, in France; but it gave dissatisfaction, as the troy pound did
not weigh so much as the pound then in use; consequently, a mean weight
was established, making 16 ounces equal to one pound, and called avoir
dupois {avoir du poids), ,
Three grains of barleycorn, welldried, placed end to end, made an inch
— the basis of length. The length of the arm of King Henry I. was made
the length of the ulna^ or ell, which answers to the modern yard. The
imperial standard yard is a solid square bar of gunmetal, kept in the
office of the Exchequer at Westminster, 38 inches in length, i inch square,
at the temperature 62° F., composed of copper 16 ounces, tin 2^ ounces,
and zinc i ounce. Two cylindrical holes are drilled half through the bar,
one near each end, and the centres of these holes are 36 inches, or 3 feet,
apart — the length of the imperial standard yard. Compared with a pendu
lum vibrating seconds of mean time, at the level of the sea, in the latitude
of London, in a vacuum, the yard is as 36 inches in length to 39.1393
inches, the length of the pendulum.
Measures of capacity were based on troy weight; it was enacted that
8 pounds troy of wheat, from the middle of the ear, well dried, should
make i gallon of wine measure, and that 8 such gallons should make
I bushel.
The imperial gallon is now the only standard measure of capacity, and it
contains 277.274 cubic inches. It is said to be the volume of 10 pounds
avoirdupois of distilled water, weighed in air, at 62** F.
Note, — The exact volume of 10 pounds of distilled water at 62** F. is
277.123 cubic inches.
GREAT BRITAIN AND IRELAND.— LENGTH. 1 29
Tables of weights and measures are conveniently classified thus —
I. Length; 2. Surface; 3. Volume; 4. Capacity; 5. Weight.
The following are some of the principal units of measurement : —
The acr^j for land measure.
The mi/^, for itinerary measure.
The yardj for measure of drapery, &c.
The coomb^ for capacity of com, &c.
The gallon^ for capacity of liquids.
The graifij for chemical analysis.
TYsQ found, for grocers* ware, &c
The stone of 8 pounds, for butchers' meat
The stone of 14 pounds, for flour, oatmeal, &c.
I. Measures of Length. — Tables No. 12.
Lineal Measure.
3 barleycorns, or'V
12 lines, or f • i.
72 pomts, or I
1000 mils /
3 inches i palm.
4 inches i hand.
9 inches v i span.
12 inches '. i foot
18 inches i cubit
3 feet I yard.
2^ feet i.military pace.
5 feet I geometrical pace.
2 yards i fathom.
5j^ yards * i rod, pole, or perch.
zt^r] ^fi«
8 furlongs, or "j
1760 yards, or > i mile.
5280 feet j
3 miles I league.
2240 yards, or ) j . , .,
The inch is also divided into halves, quarters, eighths, and sixteenths;
sometimes mto tenths.
The hand is used as a measure of the height of horses.
The miUtary face is the length of the ordinary step of a man.
The geometrical pace is the length of two steps. A thousand of such
paces were reckoned to a mile.
The fathom is used in soundings to ascertain depths, and for measuring
cordage and chains.
9
130 WEIGHTS AND MEASURES.
Land Measure,
7.92 inches i link.
100 links, or \
^^^^^\^^ \ X chain.
22 yards, or f
4 poles )
10 chains i furlong.
80 chains, or ) _.,
8 furlong / ^ °^^
The^, or woodland pole ox ferchy is 18 feet
Tht forest pole \s 21 feet. .
Nautical Measure.
6086.44 feet, or \
1000 fathoms, or ( f i nautical mile,
10 cables, or C ( or knot.
1. 1528 statute miles )
3 nautical miles i league.
60 nautical miles, or \
69.168 statute miles or > i degree.
20 leagues j
( Circumference
360 degrees...' < of the earth at
( the equator.
The above value of the nautical mile is that which is commonly taken,
and is the length of a minute of longitude at the equator. The mean
length of a minute of latitude at the mean level of the sea is nearly 6076
feet, or 1.1508 statute miles.
The nautical fathom is the thousandth part of a nautical mile, and is, on
an average, about ^th longer than the common fathom.
Cloth Measure,
2}^ inches , i nail.
2 nails I fingerlength.
4 nails, or 9 inches i quarter.
4 quarters i yard.
5 quarters i elL
WireGauges.
The " Birmingham WireGauge " is a scale of notches in the edge of a
plate, of successively increasing or decreasing widths, to designate a set of
arbitrary sizes or diameters of wire, ranging from about half an inch down to
the smallest size easily drawn, say, fourthousands of an inch. The practical
utility of such a gauge is obvious, when it is considered how far beyond the
means supplied by the graduations of an ordinary scale of feet and inches
is the measurement of the gradations of the wiregauge. But the "Birming
ham WireGauge" is a variable measure. The principle, if there was any,
on which it was originally constructed, is not known. Mr. Latimer Clark
states that, when plotted, the width? of the gauge range in a curve approxi
GREAT BRITAIN AND IRELAND. — WIREGAUQES.
131
mating to a logariAmic curve, such as would be found by the successive
addition of 10 or 12 per cent to the width of the notches of the gauge.
However that may be, there are many varieties of the wiregauge in existence.
The oldest and bestknown gauge is that of which the numbers were care
fully measured by Mr. Holtzapffel, and published by him in 1847. It has
been, and still is, widely followed in the manufacture of wire; and also of
tubes in respect of their thickness. It gives 40 measurements ranging from
.454 inch to .004 inch, and is contained in Table No. 13. Although
there are only 40 marks in the table, there are 60 different sizes of wire
made, for which intermediate sizes have been added to the gauge. This
table haJs also been used in rolling sheet iron, sheet steel, and other
materials, and for joiners' screws; but it appears to be falling into disuse
for these purposes.
Birmingham WireGauge (HoltzapffeVs), — Table No. 13.
For Wire and Tubes chiefly; and for Sheet Iron and Steel formerly.
! Maik.
SlTT,
Mark.
Si2e.
Mark.
Size.
Mark.
Size.
No.
Inch.
No.
, Inch.
No.
Inch.
No.
Inch.
0000
.454
7
.180
17
.058
27
.016
000
.425
8
.165
18
.049
28
.014
00
.380
9
.148
19
.042
29
.013
•340
10
.134
20
•035
30
.012
I
.300
II
.120
21
.032
31
.010
2
.284
12
.109
22
.028
32
.009
3
259
13
•09s
23
.025
33
.008
4
.238
14
.083
24
.022
34
.007
5
.220
15
.072
25
.020
35
.005
6
.203
16
.065
26
.018
36
.004
BiRMiNGH.^M Metai^Gauge, or PlateGauge (HoltzapffePs), —
Table No. 14.
For Sheet Metals, Brass, Gold, Silver, &c.
1 M«k.
Six.
Mark.
Sire.
Mark.
Size.
Mark.
Size.
Ko.
Inch.
No.
Inch.
No.
.Inch.
No.
Inch.
I
.004
10
.024
19
.064
28
.120
2
.005
II
.029
20
.067
29
.124
3
.008
12
.034
21
.072
30
.126
4
.010
13
.036
22
.074
31
.133
5
.012
14
.041
23
.077
32
.143
6
.013
15
.047
24
.082
33
.145
7
'OI5
16
•051
25
.095
34
.148
8
.016
17
.057
26
.103
35
.158
9
.019
18
.061
27
.113
36
.167
Another of HoltzapffeVs tables, No. 14, the PlateGauge^ has been, and
may now, to some extent, be, employed for most of the sheet metals, except
132
WEIGHTS AND MEASURES.
Lancashire Gauge {Holtzapffd' s\ — Table'^o. 15.
For Round Steel Wire, and for Pinion Wire.
Maik.
No.
Size.
Mark.
No.
Size.
Mark.
Sirr.
Mark.
No.
Size.
Maik.
Size.
Inch.
Inch.
No.
Inch.
Inch.
No.
Inch.
80
.013
57
.042
34
.109
II
.189
M
•295
79
.014
56
.044
33
.III
10
.190
N
.302
78
.015
55
.050
32
•115
9
.191
.316
77
.016
54
•055
31
.118
8
.192
P
.323
76
.018
53
.058
30
•125
7
.195
Q
•332
75
.019
52
.060
29
.134
6
.198
R
•339
74
.022
51
.064
28
.138
5
.201
S
•348
73
.023
50
.067
27
.141
4
.204
T
.358
72
.024
49
.070
26
■143
3
.209
U
.368
71
.026
48
•073
25
.146
2
.219
V
377
70
.027
47
.076
24
.148
I
.227
w
.386
69
.029
46
.078
23
.150
A
.234
X
■397
68
.030
45
.080
22
.152
B
.238
Y
.404
67
.031
44
.084
21
.157
C
.242
Z
413
66
.032
43
.086
20
.160
D
.246
Ai
.420
65
*^ZZ
42
.091
19
.164
E
.250
Bi
■431
64
.034
41
.095
18
.167
F
•257'
Ci
•443
63
•035
40
.096
17
.169
G
.261
Di
.452
62
.036
39
.098
16
.174
H
.266
Ei
.462
61
.038
38
.100
15
.175
I
.272
Fi
.475
60
.039
37
.102
14
.177
J
.277
Gi
■484
59
.040
36
•105
13
.180
K
.281
Hi
.494
58
.041
35
.107
12
.185
L
.290
ing iron and steel : as copper, brass, gildingmetal, gold, silver, and platinum.
The intervals are closer or smaller than those of the wiregauge, and the
maximum size, for No. 36, is '/6 inch. When thicker sheets are wanted,
their measures are sought in the Birmingham wiregauge.
The last table, No. 15, by Holtzapffel, the Lancashire Gauge, is employed
exclusively for the bright steel wire prepared in Lancashire, and the steel
pinionwire for watch and clock makers. The larger sizes are marked by
capital letters, to distinguish them from the others. This, the second part
of the table, is known as the LetterGauge,
Needle Gauge, for needle wire. The sizes correspond with some of those
of the Holtzapffel wiregauge. The following are the relative marks for
equal sizes on the two gauges : —
Needle wire gauge — Nos. i, 2, 2j^, 3, 4, 5, thence to 21,
corresponding to B. W.G. — i8j^, 19, 19^, 20, 21, 22, thence to 38.
* Music IViregauge, for the strings of pianofortes. The marks used are
Nos. 6 to 20. The following are the relative marks for equal sizes with the
Holtzapffel wiregauge: —
Music wiregauge — Nos. 6, 7, 8, 9, 10,11, 12,14,16,18,20,
corresponding to B. W.G. — 26, 25)^, 25, 24}^, 24, 23 J^, 23, 22, 21, 20, 19.
No. 6, the thinnest wire now used, measures about one fiftyfifth of an inch
in diameter, and No. 20 about one twentyfifth of an inch.
GREAT BRITAIN AND IRELAND. — WIREGAUGES.
133
The preceding Tables of Gauges have been extracted from HoltzapfFers
estimable work on Turning and Mechanical Majnipulation^ 1847.
Messrs. Rylands Brothers, of Warrington, manufacture iron wire accord
iDg to the gauge in Table No. 16.
Warrington WireGauge (Rylands Brothers), — Table No. 16.
Mark.
Size.
Mark.
Size.
Mark.
Size.
Mark.
No.
Size.
No.
Inch.
No.
Inch.
No.
Inch.
Inch.
7/0
1/2
.326
8
•159
15
.069
6/0
15/32
I
.300
9
.146
16
.0625, or Vx6
5/0
7/16
2
.274
TO
•133
17
.053
4,^
13/32
3
25, or K
io>^
.125, or^
18
.047
3/0
3/8
4
.229
II
.117
19
.041
2/0
11/32
5
.209
12
.io,or V,o
20
.036
6
.191
13
.090
21
•o3i5»orV33
7
•174
14
•079
22
.028
For sheets, the wiregauge that seems to be adhered to by the ironsheet
rollers of South Staffordshire, is a scale comprising 32 measurements, ranging
from .3125 inch to .0125 inch, contained in Table No. 17.
Birmingham WireGauge. — ^Table No. 17.
For Iron Sheets chiefly.
No.
Size.
No.
Inch.
3125 (Vx6)
9
28125
10
25 (X)
II
234375
12
21875
13
203125
14
1875 (V,6)
15
171875
16
Size.
No.
Inch.
•15625
.140625
.125 (>^)
17
18
19
.1125
20
•10 (Vio)
.0875
21
22
.075
.0625 (Vxe)
23
24
Size.
No.
Inch.
.05625
25
.05 (Vo)
26
•04375
27
.0375
28
•034375
29
.03125(732)
30
.028125
31
•025 (V40)
32
Inch.
.02344
.021875
.020312
.01875
.01719
.015625
.01406
.0125 (Veo)
Sir Joseph Whitworth, in 1857, introduced his Standard WireGauge,
ranging fix)m a half inch to a thousandth of an inch, and comprising 62
measurements, as given in Table No. 18. It commences with the
smallest size, and increases by thousandths of an inch up to half an inch.
The smallest size, V'loooth of an inch, is No. i ; No. 2 is Vioooths of an inch,
and so on, increasing up to No. 20 by intervals of Vioooth of an inch; from
No. 20 to No. 40 by '/loooths; from No. 40 to No. 100 by s/ioooths of an
inch. The sizes are designated or marked by their respective numbers in
thousandths of an inch.
It appears that the Whitworth Gauge is entering into general use ; and,
in the manufacture of wire, at least, this and Rylands* gauge are likely soon
to supersede the Holtzapifel scale.
134
WEIGHTS AND MEASURES.
Sir Joseph Whitworth & Co.'s Standard WireGauge. — ^Table No. i8.
Mark.
Size.
Mark.
Size.
Mark.
Size.
Mark.
Size.
No.
Inch.
No.
Inch.
No.
Inch.
No.
Inch.
I
.001
17
.017
55
.055
200
.200
2
.002
18
.018
60
.060
220
.220
3
.003
19
.019
6$
.065
240
.240
4
.004
20
.020
70
.070
260
.260
S
.005
22
.022
75
.075
280
.280
6
.006
24
.024 •
80
.080
300
.300
7
.007
26
.026
85
.085
325
.325
8
.008
28
.028
90
.090
350
.350
9
.009
30
.030
95
.095
375
•375
ID
.010
32
.032
100
.100
400
.400
II
.Oil
34
.034
no
.110
425
.425
12
.012
36
.036
120
.120
45«>
.450
13
.013
3S
.038
135
•135
475
.475
14
.014
40
.040
150
.150
500
.500
15
.015
45
.045
165
.165
16
.016
50
.050
180
.180
Common Fractions of an Inch and Holtzapffel's WireGauge. —
Table No. 19.
Fraction.
Inch.
'A
'A
■A
'A
'A
•A
■A
'Ac
'A.
•A,
'/.6
•/
:^
WireGauge.
so
21
a«
24
25
Va6
No.
fuU
bare
bare
bare
full
full
full
bare
o
3
6
8
9
I
2
3
3
4
4 bare
5 rather bare
6
6 bare
7fuU
7 bare
8 full
8 rather full
8 rather bare
8 bare
9 rather full
9
9 bare
20 full
Fraction.
Inch.
Va8
V3«
V37
V38
V39
'Ac
v::
'r
Vs6
V58
V60
WireGauge.
No.
20 rather full
20
20 rather bare
21 rather full
21
21 rather bare
21 bare
22 full
22 rather full
22
22 bare
23 full
23 rather full
23
23 bare
24 full
24
25fuU
25
25 bare
26 rather full
26
26 bare
27 full
Fraction.
WireGauge.
Inch.
V
70
V74
V76
'As
Vso
•As
:^
/95
V
:;■
Vx3o
V140
Vis©
VI60
180
100
zo
xao
V
7
aoo
950
No.
27
27 rather
27 bare
28 full
28 rather
28
28 bare
29 rather
29
29 bare
30 rather
30 bare
31 rather
31
32
33 full
33 l>are
34
34 rather
34 bare
35 Ml
35
36
bare
full
full
full
full
bare
GREAT BRITAIN AND IRELAND. — FRACTIONS OF INCH. 1 35
Inches and their Equivalent Decimal Values in Parts of a Foot.
— Table No. 20.
Inches.
Fiactionoffoot
Foot.
I
Via
08^1
2
7
:;
.1667
25
4
c
1
•3333
4167
D
6
7
■A
7/„
...... .Af * vr f
■5
eSiX
/
8
/la
Va
3/.
oo
.6667
7c
V
10
II
/o
.8333
QI67
12
•
/ xa
I
y *^/
I.O
Fractional Parts of an Inch, and their Decimal Equivalents.
Tables No. 21.
Eighths.
Eighths.
Fractions.
Inch.
I
i/g
12^
2
X
/o
•A
S/g
^•J
77c
4
c
/<»
V.
S/g
o# J
•5
621:
6
7
/«»
7/.
•*'• J
•75
87s
/ ••••••
8
I
/ J
1.0
Twelfths,
Twelfths.
Fractions.
Inch.
I......
«/„
oZx%%
2
'A
i/«
•125
16667
3
4.
'A
V,
•^5
'?'?^^'?
^
5
6.
'k
•• 00000
.41667
5
7
8
/*
t
•5^333
66666
9
10
5/g
•75
S'?^^^
II
12
/o
"A.
I
^0000
.91667
1.0
136 WEIGHTS AND MEASURES.
Sixteenths and Thirtyseconds. — Tables No. 21 (continued).
Thirty
Seconds.
Sutteenths.
Fractions.
Inch.
I
2
3
4
5
6
7
8
9
10
II
12
13
14
IS
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
I
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
v„ .:
.03125
.0625
•0937s
"5
•15625
.1875
.21875
•*S
.28125
.3125
•34375
•375
.40625
•4375
.46875
•S
•53125
•5625
•59375
.625
.65625
•6875
•71875
•75
.78125
.8125
.84375
.875
.90625
•9375
.96875
1.0
/3*
3/,.
/ 3'
5/„
'3a •
7/„
/3«
V, :'•....
/3*
3/3
»3/„
/3a
»/. !{:!...
»7/„
/ 3*
'A6
«9/^
/3« •
91/
/ 3*
"A6
23/„
/ 3«
»s/..
'3« •
a7/„
/ 3*
ap/,^
'3* •
'V.6
3x/„
/ 3*
I
II. Measures of Surface. — Tables No. 22.
Superficial Measure,
144 square inches, or 7 
183.35 circular inches 3 i square foot
9 square feet i square yard
loo square feet i square.
27 2 J^ square feet, or ) nH
30^ square yards j ' ^^*
The square is used in measuring flooring and roofing.
The rod is used in measuring brickwork.
GREAT BRITAIN AND IRELAND. — SURFACE, VOLUME. 1 37
Builder^ Measurement,
I superficial part i square inch.
12 parts "i inch" (12 square inches).
12 "inches" i square foot.
This table is employed in the superficial or flat measure of boards, glass,
stone, artificers' work, &c.
Land Measure.
9 square feet i square yard.
30X square yards j ^ ^^l^^
16 square poles i square chain.
40 square poles, or ) ^
1 2 10 square yards J
4 roods, or
10 square chains, or
160 square poles, or \ i acre.*
4,840 square yards, or
43,560 square feet
640 acres, or ) 1
3,097,600 squa^ yards } i square imle.
30 acres i yard of land.
100 acres i hide of land.
40 hides I barony.
* The side of a square having an area of one acre is eqnal to 69.57 lineal yards.
III. Measures of Volume. — Tables No. 24.
Solid or Cubic Measure,
1728 cubic inches \
2200.15 cylindrical inches ( i cubic foot
3300.23 spherical inches j
6600.45 conical inches )
27 cubic feet i cubic yard, or load.
35.3156 cubic feet or \ ^ ^^^j^
1.308 cubic yards J
^ate. — ^The numbers of cylindrical, spherical, and conical inches in a cubic foot, are
as I, 1. 5, 3
Builderi Measurement,
I solid part 12 cubic inches.
12 solid parts i "inch" (144 cubic inches).
12 "inches" i cubic foot
This table is used in measuring squaresided timber, stone, &c.
138
WEIGHTS AND MEASURES.
Note, — ^The cubic contents of a piece,
6 inches square and 4 feet long is i cubic foot
7
8J^
12
17
24
3
2
I
I
I
»
I
I
I
2
4
Decimal Parts of a Square FogTj
, IN Square Inches. — Table No. 23.
Hundredth
Square
Hundredth
Square
Hundredth
Square
Hundredth
Square
Parts.
Inches.
Parts.
Ihches.
Parts.
Inches.
Parts.
Inches.
I
1.44
26
37.4
51
73.4
76
109.4
2
2.88
27
38.9
52
74.9
77
IIO.9
3
4.32
28
40.3
53 •
76.3
78
II2.3
4
576
29
41.8
54
77.8
79
1 13.8
5
7.20
30
43.2
55
79.2
80
II5.2
6
8.64
31
44.6
56
80.6
81
I16.6
7
10. 1
32
46.1
57
82.1
82
1 18. 1
8
"5
33
47.5
58
83.5
83
"95
9
13.0
34
49.0
59
85.0
84
121.0
10
14.4
35
504
60
86:4
85
122.4
II
iS8
36
51.8
61
87.8
86
123.8
12
17.3
37
533
62
89.3
87
125.3
13
18.7
38
54.7
63
90.7
88
126.7
14
20.2
39
56.2
64
92.2
89
128.2
. 15
21.6
40
57.6
65
936
90
129.6
16
23.0
41
58.0
66
95.0
91
131.0
17
24.5
42
60.5
67
96.5
92
132.5
18
259
43
61.9
68
97.9
93
133.9
19
27.4
44
63.4
69
994
94
1354
20
28.8
45
64.8
70
100.8
95
136.8
21
30.2
46
66.2
71
102.2
96
138.2
22
317
47
67.7
72
103.7
97
1397
23
33.1
48
69.1
73
105.1
98.
141. 1
24
34.6
49
70.6
74
106.6
99
142.6
25
36.0
50
72.0
75
108.0
100 144.0
IV. Measures of Capacity. — Tables No. 25.
Liquid Measure,
8.665 cubic inches i gill or quartern.
4 gills (34.659 cubic inches) i pint
2 pints I quart
2 quarts i pottle.
4 quarts, or 8 pints (277.274 cubic inches) i gallon.
6.2355 gallons I cubic foot
The hamgallon^ for milk, is equal to 2 imperial gallons.
GREAT BRITAIN AND IRELAND. — CAPACITY. 1 39
Dry Measure,
2 pints I quart
4quarts i gallon.
2 gallons I peck.
8S;s°'} (I 28366 cubic feet) i bushel
2 bushels I strike.
4bushels i coomb.
5 bushels I sack.
8 bushels i quarter.
4 quarters (41.Q77 cubic feet) i chaldron.
5quarters i wey or load.
2 loads I last
The standard bushel is 18^ inches in diameter inside, and 8^ inches
deep; it holds 80 pounds of distilled water at 62'' F. It is 19^^ inches in
diameter outside. This measure is applied to dry goods, as com, seeds,
roots, &C., which are struck with a straight bar. The old dry measures had
the same denominations and proportions, and were 96.95 per cent of the
Imperial Dry Measures, above given.
The heaped imperial bushel must be an upright cylinder, of which the
diameter is not less than twice its depth, and the height of the conical heap
must be at least threefourths of the depth of the bushel, the outside of the
pleasure being the boundary of the base of the cone. It may be 18.789
inches in diameter inside, and 8 inches deep; and the capacity, heaped,
must be 1.6293 cubic feet. Heaped measure is used for such goods as
camiot be conveniently stricken, as coals, potatoes, and fruit
Coal Measure,
3 bushels (heaped) i sack.
9 bushels I vat
36 bushels, or 1 2 sacks i chaldron.
5^ chaldrons i room.
21 chaldrons i score.
Old Wine and Spirit Measure,
•11 \ Imperial
4 g^lS I pint Gafions.
2 pints I quart
4 quarts (231 cubic inches) i gallon = .8333
10 ^lons ranker = 8.333
18 gallons I runlet = 15.
31}^ gallons I barrel = 26.250
42 gallons I tierce = 35.
'jgS"'} hogshead =5.5
.^*CS^°J} , p»,cheo„ . 70. ;
126 gallons, or \
2 hogsheads, or > i pipeorbutt= 105.
i^ puncheons )
2 pipes, or ) ^ .^ ^,r.
*^ '^ i > I tun =210.
3 puncheons j
I40 WEIGHTS AND MEASURES.
By this measure wines, spirits, cider, perry, mead, vinegar, oil, &c., are
measured; but the contents of every cask are reckoned in imperial gallons
when sold. The imperial gallon is onefifth laiger than the old wine
gallon.
Old Ak and Beer Measure,
2 pints I quart JSESS!*
4 quarts (282 cubic inches) i gallon = 1.017
9 gallons I firkin = 9.153
2 firkins, or 18 gallons 1 kilderkin = 18.306
2 kilderkins, or I , u^^^i r a
, 11 ' > I barrel = 36.612
36 gallons J ^
zoiX'"} ^''"" =^°^«^^
The imperial gallon is onesixtieth smaller than the old beer gallon.
Apothecaries Fluid Measure,
60 minims (ni) i fluid drachm (/5).
8 drachms (water, 1.732 cubic) « • , / ^ ^x
inches, is? >^ grains) } ^ ^^'^ ^"""^ (/ D
20 ounces i pint ( ^ )•
8 pints (water, 70,000 grains) i gallon {i^^)
1 drop I grain.
60 drops I drachm.
4drachms i tablespoonfiiL
2 ounces (water, 875 grains) i wineglassful.
3 ounces i teacupful.
V. Measures of Weight. — Tables No. 26.
Avoirdupois Weight,
16 drachms, or I / v
,^^1/ ,^:„« \ I ounce(^2:.).
437?^ grams j • ^ '
16 ounces, or ) j /• • i\ //z \
7000 grains } ^ P^""*' (""Penal) {ib.).
8 pounds I stone (London meat market).
14 pounds I stone.
28 pounds, or I ^ / \
2 stones I ' quarter (^r.).
4 quarters, or \
8 stones, or > i hundredweight {cwt).
112 pounds j
20 hundredweights i ton.
The grain above noted, of which there are 7000 to the pound avoirdupois,
is the same as the troy grain, of which there are 5760 to the troy pound.
Hence the troy pound is to the avoirdupois pound as i to 1.2 15, or as
14 to 17.
\
GREAT BRITAIN AND IRELAND.— WEIGHTS. I41
The troy ounce is to the avoirdupois ounce as 480 grains, the weight of
the former, to 43 7 >^ grains, the weight of the latter; or, as 1 to .9115.
In Wales, the iron ton is 20 cwt of 120 lbs. each.
Troy Weight.
24 grams i pennyweight (^/o^/.).
20 pennyweights, or )
480 grains / ' ^^"^^•
1 2 ounces, or ) •
5760 grains / ^ P°"°<^
25 pounds I quarter.
4 quarters, or loo poUnds i hundredweight.
By troy weight are weighed gold, silver, jewels, and such liquors as are
sold by weight
Diamond Weight
I diamond grain 0.8 troy grain.
I carat 4 diamond grains.
15J4 carats i troy ounce.
Apothecaries Weight.
The revised table of weights of the British Pharmacopeia is as follows :
it is according to the avoirdupois scale : —
4375^ grains.... i ounce.
16 ounces i pound.
In the old table of Apothecaries* Weight, superseded by the preceding
table, the troy scale was followed, thus: —
Old Apothecaries Weight.
20 grains i scruple O).
3 scruples, or ) j 1. /\
6ograi^ I I drachm (3).
8 drachms, or ) ' /«\
480 grains } ^ °"'»^« <?)•
12 ounces, or ) j /7il\
5760 grains } • ^ POund (/*.)•
Weights of Current Coins.
I farthing, .8 inch diameter, »/,o ounce.
I halfpenny, i.o „ Yj „
I penny, 1.2 „ V3 »
I threepenny piece »/«> ^i
I fourpenny piece '/xs „
I sixpence Yio „
I shilling V5 „
I florin »/s »
I halfcrown Ya „
5 shillings or 10 sixpences i „
I sovereign »/^ ounce, nearly.
For the exact weight in grains of these coins, see Table of British Money.
142 WEIGHTS AND MEASURES.
Coal Weight.
14 pounds I stone.
28 pounds I quarter hundrpdweighL
56 pounds I half hundredweight.
88 pounds i bushel.*
I sack, of 1 1 2 pounds i hundredweight
I double sack, of 224 pounds... 2 hundredweights.
20 hundredweights, or I ,
10 double sacks J
26j^ hundredweights i chaldron (London).
53 hundredweights i chaldron ( Newcasde).
7 tons T room.
21 tons 4 cwt I barge or keel.
* Sundry Bushel Measures,
I Cornish bushel of coal is 90 or 94 pounds ;. heaped, loi pounds.
I Welsh bushel, average wei£[ht 93 pounds.
I Newcastle bushel is 80 or 84 pounids. Bradley Main, 92^ pounds.
I London bushel, 80 or 84 pounds.
f In Wales the miners* coalton is 21 cwt. of 120 lbs. each.
Wool Weight
' 7 pounds I clove.
2 cloves, or 14 pounds i stone.
2 stones I tod.
6]4 tods I wey.
2 weys I sack.
12 sacks, or 39 hundredweight i last.
12 score, or 240 pounds i pack.
Hay and Straw Weig/U.
I truss of straw 36 pounds.
I load of straw 11 hundredweights, 64 pounds.
I truss of old hay 56 pounds.
I load of old hay 18 hundredweight.
I cubic yard of old hay 9 stone.
I cubic yard of oldish hay 8 stone.
I truss of new hay 60 pounds.
1 load of new hay ; 19 hundredweights, 3 2 pounds.
I cubic yard of new hay 6 stone.
Com and Flour Weight
1 peck, or stone of flour 14 pounds.
10 pecks I boll = 140
2 bolls I sack =280
14 pecks I barrel =196
I bushel of wheat 60
I bushel of barley ^ 47
I bushel of oats 40
Six bushels of wheat should yield one sack of flour; i last of corn is 80 bushel&
GREAT BRITAIN AND IRELAND.— MISCELLANEOUS. I43
Miscellaneous Tables. — No. 27.
Whatman* s Drawing Papers, — Sizes of Sheets,
Antiquarian 53 inches long, 3 1 inches wide.
Doubleelephant 40
Atlas 34
Colombier 34
Imperial 30
Elephant 28
Superroyal 27
Royal 23
Medium 22
Demy 20
n
27
26
23
22
23
19
19
17
15
Commercial Numbers and Stationery.
12 articles 1 dozen.
13 articles i long dozen.
12 dozen i gross.
20 articles i score.
5 score I common hundred.
6 score i great hundred.
30 deals I quarter.
4 quarters i hundred.
24 sheets of paper i quire.
20 quires i ream.
2ij quires i printers* ream.
5 dozen skins of parchment i roll.
Measures relating to Building,
Load of timber, unhewn or rough 40 cubic feet
Load, hewn or squared [ 5° cubic feet, reckoned
* ^ (to weigh 20 cwt
Stack of wood io8 cubic feet.
Cord of wood 128 „
(In dockyards, 40 cubic feet of hewn timber are reckoned to weigh
20 cwt. ; 50 cubic feet is a load.)
100 superficial feet i square.
Himdred of deals 120 deals.
Load of iinch plank 600 square feet.
(Load of plank more than iinch thick = 600 ^ thickness in inches.
Planks, section 11 by 3 inches.
Deals, section 9 by 3 „
Battens, section 7 by 2^ „
A reduced deal is 1 J^ inches thick, 1 1 inches wide, and 1 2 feet long.
Bundle of 4 feet oakheart laths 120 laths.
Load of „ „ 3 7 J^ bundles.
Bundle of 5 feet oakheart laths 100 laths.
Load of „ „ 30 bundles.
144 WEIGHTS AND MEASURES.
Measures reiaiing to Building {continued,)
Load of statute bricks 500.
Load of plain tiles 1000,
Load of lime 32 bushels.
Load of sand 36 „
Hundred of lime 35 „
Hundred of nails, or tacks 120.
Thousand of nails, or tacks 1 200.
Fodder of lead iqJ^ cwt.
Sheet lead 6 to 10 pounds per sq, ft
Hundred of lead 112 pounds.
Table of glass 5 feet.
Case of glass 45 tables.
case of glass { ^^^"^^f taS~''
Stone of glass 5 pounds.
Seam of glass 24 stone.
Sundry Commercial Measures,
Dicker of hides ; 10 skins.
Last of hides 20 dickers.
Weigh of cheese 256 pounds.
Barrel of herrings 26 V3 gallons.
Cran of herrings 37^ „
Pocket of hops i ^ to 2 cwt.
Bag of hops 3j^ cwt, nearly.
Last of potash, codfish, white her ) barrels.
rings, meal, pitch, tar j
Barrel of tar 26 J4 gallons.
Barrel of anchovies 30 pounds.
Barrel of butter 224 „
Barrel of candles 120 „
Barrel of turpentine 2 to 2ji c^vt.
"Barrel of gunpowder 100 pounds.
Last of gunpowder 24 barrels.
Measures for Ships,
I ton, displacement of a ship, 35 cubic feet
I ton, registered internal capacity of do., 1 00 do.
I ton, shipbuilders' old measurement, 94 do.
Comparison of Compound Units. — ^Tables No. 28.
Measures of Velocity,
^y ^^ji. S ^•467 feet per second.
I mile per hour { 88.0 feet per minute.
I knot per hour i,688 feet per second.
I foot per second .682 mile per hour.
I foot per minute .01136 mile per hour.
GREAT BRITAIN AND IRELAND. — COMPOUND UNITS. 14$
Measura of Volume and Time,
I cubic foot per second [ ^""^ ^"^!^ ^^^^ P^^' T'"'*^
I ^33333 cubic yards per hour.
I cubic foot per minute 2.222 cubic yards per. hour.
I cubic yard per hour .45 cubic foot per minute.
I cubic inch per second [ ''•^^ cubic foot per hour.
^ ( 12.984 gallons per hour.
I gaUon per second 569. 124 cubic feet per hour.
I gallon per min ute 9485 cubic feet per .hour.
Measures of Pressure and Weight (See also page 127.)
i 144 lbs. per square foot.
I lb. per square inch < 1296 lbs. per square yard.
i 57^6 ton per square yar^.
1 atmosphere (14.7 lbs.) per ) g ^^^ ^^
square mch j ^ ^ v ^ j
i .00694 lb. per square inch.
I lb. per square foot < .11 ii ounce per square inch.
( .0804 cwt per square yard.
,, . , ( 2.0355 inches of mercury at 32° F.
1 lb. per square inch ^ ^.j^^^f^et of water at s l\z F.
. , X. 1. o c f .401 lb. per square inch.
I inch of mercury at 32 F. j ^ ^^^ feet of water at S2".3 F.
( 4333 lb. per square inch.
I foot of water, at S2°.3 F. .. < 62.4 lbs. per square foot.
( .8823 inch of mercury at 32® F.
Measures of Weight and Volume,
{405. 1 grains per cubic inch.
.926 ounce per cubic inch.
4.107 cwt. per cubic yard.
1.205 tons per cubic yard.
, . . , f ^•O'^o ounces per cubic foot.
1 grain per cubic mch { ^ ^J^ pounds per cubic foot
I ounce per cubic inch 1 08 pounds per cubic foot.
1 cwt. per cubic yard 4. 1 48 pounds per cubic foot.
I ton per cubic yard 8 2. 963 pounds per cubic foot
r I pound for 1 1 2 2 cubic feet.
I grain per gallon (i in 70,000
parts by weight, of water)
I pound for 41.5 cubic yards.
I pound for 31.8 cubic metres.
220 grains for i cubic metre.
.503 ounce for i cubic metre.
Measures of Power,
w. ce \ xj T> ( 1,980,000 footpounds per lb. of fuel.
1 lb. of tuei per n.i'. \ ^21.76 million footpounds per cwt of fuel
P^^ ^^"^ i 2,565 units of heat
i,ooo,ooofoo^p^^^^ I j^3 p^^^^g ^^f^^i p^^ H.P. per hour.
146 WEIGHTS AND MEASURES.
FRANCE.— THE METRIC STANDARDS OF WEIGHTS
AND MEASURES.
The primary metric standards are : — the metre, the unit of length ; and
the kilogramme, the unit of weight, derived from the metre : being the two
platinum standards deposited at the Palais des Archives at Paris.
The standard metre is defined to be equal to one tenmillionth part of
the quadrant of the terrestrial meridian, that is to say, the distance from
the equator to the pole, passing through Paris, which, by the latest and
most authoritative measurement, is 39.3762 inches, in terms of the Imperial
standard at 62° F. By the latest and most accurate measurement, the
actual standard metre at 32° F. is, in terms of the Imperial standard at 62° F.,
39.37043 inches; and its legal equivalent, declared in the Metric Act of
1864, is ^9.3708 inches, being the same as that adopted in France.
The standard kilogramme (looo grammes) is defined to be the weight of
a cubic decimetre of distilled water at its maximum density, at 4^0 C.
or 39°. I F. This is legally taken to be
.2.20462125 lbs., or
, 2 lbs., 3 oz., 4.383 drachms, or
i5»432.34874 grains.
There is in the Standard Department at Westminster a newlyconstructed
subdivided standard yard, laid down upon a bar of Baily's metal, upon
which a subdivided metre has also been laid down.
The metric unit of capacity is the litre, defined to be equal to a cubic
decimetre. Its Imperial equivalent is 0.22009 gallon.
There is no other official standard of weight and measure in France
than the metre and the kilogramme; there is no standard litre or unit of
capacity.
The metric system is not really founded on the length of a quadrant of the
meridian, and although it is described as a scientific system, because of the
simple and definite relation between the metre, which is its basis and unit of
length, and the kilogramme and litre, which are the units of weight and
capacity, it is admitted that it has been found impossible practically to
carry it out with scientific accuracy. The standard kilogramme is admitted
not to be actually the weight of a cubic decimetre of pure water at the
specified temperature, nor the litre a measure of capacity holding a cubic
decimetre of pure water. The real standard unit of weight is declared, even
by men of science in France, to be merely the platinum kilogrammeweight
deposited at the Palais des Archives, as the real standard unit and basis of
the metric system is the platinum metre, also deposited there. It is an
accomplished fact, however, that all civilized nations have tacitly agreed to
recognize the metric system as affording for the future the advantages of a
universal system of weights and measures, and to adopt the standards
deposited at the Palais des Archives as the primary units of the system.
The French metric system has been adopted, and its use made compul
sory by the following States: — France and Belgium, in 1801; Holland, in
1819; Greece, in 1836; Italy and Spain, in 1859; Portugal, in 186068;
the German Empire, in 1872; Colombia, Venezuela, in 1872; Ecuador,
FRANCE.— THE METRIC STANDARDS — LENGTH. I47
Bcazfl, Peru, and Chili, in i860; also by the Argentine Confederation, and
Uruguay.
Great Britain and Ireland, in 1864, adopted the metric system, so far as to
render contracts in terms of the French metric S3rstem permissive.
The United States of North America, in 1866, legalized the French metric
system concurrently with the old system; it was also legalized in British
North America.
Switzerland, in 1856, legalized the foot of three decimetres as the unit of
length, with a decimal scale; the unit of weight being the pound of 500
grammes, or half a kilogramme, with two distinct scales of multiples and
parts, one decimal, the other according to the old custom.
Sweden, in 1855, by a law made compulsory in 1858, adopted a decimal
system of weights and measures, having for the unit of length a foot of 0.297
metre, and the unit of weight a pound of 0.42 kilogramme: — ^being the
original units decimally treated.
Denmark adopted the metric system so far as the pound of 500 grammes.
The pound is decimally treated, and since 1863 the use of the greatest
parts of the multiples of the pound not conformable to decimal sub
division has been prohibited.
Austria, in 1853, adopted a pound of 500 grammes, with decimal divisions,
for customs and ^scaJ purposes.
Russia awaits the example of those countries with which she has
conunercial relations, especially of England.
In Morocco and Tunis, the weights and measures have no relation with
the metric system.
On the 20th May, 1875, the international convention for the adoption
of the French metrical system of weights and measures was signed at Paris
by the plenipotentiaries of France, Austria, Germany, Italy, Russia, Spain,
Portugal, Turkey, Switzerland, Belgium, Sweden, Denmark, the United
States, the Argentine Republic, Peru, and Brazil. A special clause
reserves to States not included in the above list the right of eventually
adhering to the convention.
I. French Measures of Length. — Table No. 29.
I millimetre 10 centimetres.
10 centimetres i decimetre.
10 decimetres, or ]
100 centimetres, or > i metre.
1000 millimetres j
10 metres i decametre.
10 decametres x i hectometre.
10 hectometres, or 1000 metres i kilometre {kilo,)
10 kilometres i myriametre.
I toise (old measure) =1.949 metres.
1000 toises I mille = 1.949 kilometres.
2000 toises I itinerary league =3.898 „
2280.329 toises I terrestrial league =4.444 „
2850.411 toises I nautical league =5555 n
X noeud (British nautical mile) = 1.855 „
148
WEIGHTS AND MEASURES.
French WireGauges {Jauges de Fils de Fer),
The French wiregauge, like the English, has been subject to variation.
Table No. 30 contains the values of the "points," or numbers, of the
Limoges gauge; table No. 31 gives the values of a wiregauge used in the
manufacture of galvanized iron; and table No. 32 the values of a gauge
which comprises wire and bars up to a decimetre in diameter.
French WireGauge i
[J<^uge de Limoges)
.—Table No. 30.
Number.
Diameter.
Number.
Diameter. •
Number.
Diameter.
MUUmetreJ
Inch.
Millimetre.
Inch
Millimetre.
Inch.
•39
.0154
9
1.35
.0532
18
3 40
.134
I
.45
.0177
10
1.46
•0575
19
395
.156
2
.56
.0221
II
1.68
.0661
20
450
.177
3
.67
.0264
12
1.80
.0706
21
510
.201
4
.79
.0311
13
1.91
.0752
22
565
.222
5'
.90
•0354
14
2.02
•0795
23
6.20
.244
6
1. 01
.0398
15
2.14
.0843
24
6.80
.268
7
1. 12
.0441
16
2.25
.0886
8
1.24
.0488
1
17
2.84
.112
French WireGauge for Galvanized Iron Wire. — ^Table No. 31.
Number.
Diameter.
Number.
Diameter.
Number.
Diameter.
M'metre.
Inch.
M'metre.
Inch.
M'metre.
Inch.
I
.6
.0236
9
1.4
.0551
17
30
.118
2
.7
.0276
10
15
.0591
18
34
.134
3
.8
•0315
II
1.6
.0630
19
3.9
.154
4
•9
•0354
12
1.8
.0709
20
4.4
.173
5
I.O
•0394
13
2.0
.0787
21
4.9
.193
6
I.I
.0433
14
2.2
.0866
22
54
.213
7
1.2
.0473
15
2.4
.0945
23
59
.232
8
1.3
.0512
16
2.7
.106
French WireGauge. — Table No. 32.
Mark.
Size.
Mark.
Size.
Mark.
Size.
Mark.
Size.
Millimetre.
Millimetre.
Millimetre.
Millimetre.
P
5
8
13
16
27
24
64
I
6
9
14
17
30
25
70
2
7
10
IS
18
34 .
26
76
3
8
II
16
19
39
27
83
4
9
12
18
20
44
28
88
5
10
13
20
21
49
29
94
6
II
14
22
22
54
30
100
7
12
15
24
23
59
FRANCE. — THE METRIC STANDARDS. I49
II. French Measures of Surface. — Table No. 33.
1 00 square millimetres i square centimetre.
1 00 square centimetres i square decimetre.
100 square decimetres, or )
^ ^. ^ ' > I square metre, or centiare.
r 0,000 square centimetres / ^ ^ '^ > ^ ^
100 square metres, or centiares... i square decametre, or are.
100 square decametres, or ares ... i square hectometre, or hectare.
100 square hectometres, or hectares 1 square myriametre.
liand is measured in terms of the centiare^ the are, and the hectare or
arpent metrique {metric acre). There is also the decare, of 10 ares.
III. French Measures of Volume. — Tables No. 34.
Cubic Measure.
1 000 cubic millimetres i cubic decimetre.
1000 cubic decimetres i cubic metre.
Wood Measure,
10 decist^res i stfere* (i cubic metre).
I voie (Paris) 2 stferes.
I voie de charbon (charcoal) 0.2 stfere ( 75 cubic metre).
I corde 4 stferes.
* The stire measures 1. 14 metres x 0.88 metre x i metre, the billets of wood being
1. 14 metre in length.
IV. French Measures of Capacity. — ^Tables No. 35.
Liquid Measure,
10 centilitres i decilitre.
10 d^cihtres i litre.
10 litres I decalitre.
Dry Measure.
10 litres I decalitre.
10 decalitres, or) ^ hectoUtre.
100 litres J
10^ liS"*'^' °'} ' '^""^^ <' ^^'*= ™*''^>
The use of measures equal to a doublelitre^ a halflitre, a doubledicilitre^ a
halfd£cilitre^ is sanctioned by law.
ISO
WEIGHTS AND MEASURES.
V. French Measures of Weight. — Table No. 36.
10 milligrammes i centigramme.
10 centigrammes i decigramme.
10 decigrammes i gramme.
10 grammes i decagramme.
10 decagrammes i hectogramme.
10 hectogrammes, or ) ^ kilogramme (M., kUc^.)
1000 grammes J \ > c /
10 kilogrammes i myriagramme.
10 myriagtammes, or ) ^ j^ ^ ^^
100 kilogrammes J ^ ^
10 quintaux, or ) ( i millier, tonneau de mer, or tonne
1 000 kilogrammes j ( (weight of i cubic metre of water at 39°. i ).
EQUIVALENTS OF BRITISH IMPERIAL AND FRENCH METRIC
WEIGHTS AND MEASURES.
I. Measures of Length. — ^Tables No. 37.
A DBCIMBTRB DIVIDBD INTO CBNTIMBTRBS AND MILLIMBTRBS.
2
a
llljl.nlllllllHIllllliJhllllini.ll.limlhTT
llllllill
JTTTT
'mill. I
'"iImii
iiimm
UL
It I I I I I I I I 11 M l
IIMII erg:
MM
' ' ' '
I I I .1 i!
INCHBS AND TBNTHS.
Mbtsic Dbnominations
AND VaLUBS.
•
Equivalbnts in Impbrial Dbnominations.
Metres.
Inches.
Feet.
Yards.
«
MUes.
I millimetre
I centimetre
I decimetre
I M£1K£ «•..
I dekametre
I hectometre
I KILOMETRE
I myriametre
/xooo
/lOO
V.O
I
10
100
1,000
10,000
= 003937
= 0.39370
= 393704
= 3937043
= 3.28087
= 32.80869
= 3280.87
= 1.09362
10.93623
109.36231
= 1,093.6231
= 10,936.231
= 0.62138
= 6.21377
IMPERIAL AND METRIC EQUIVALENTS.
IS!
Tables No. 37 {continued).
Impbrial Denominations.
Equivalents in Metric Denominations.
Centunetres.
Metres.
Kilometres.
1
I inch I2K.A millimetres)
= 2.5399s
= , 0.30480
0.91439
1.82878
5.02915
= 20.11662
= 201.1662
= 1,609.3296
= 0.20117
= 1.60933
I foot, or 12 inches
I yard, or 3 feet, or 36 inches....
I fathom, or 2 yards, or 6 feet....
I pole, OT K'^A yards
I chain, or 4 poles, or 22 yards...
I furlong, 40 poles, or 220 yards
I mile, 8 furlongs, or 1760 yards
Equivalent Values of Millimetres and Inches. — ^Tables No. 38.
Millimetres = Inches.
MiOimetRs.
Inches.
MUiimetres.
Inches.
Millimetres.
Inches.
Millimetres.
Inches.
I
.0394
27
1.0630
53
2.0866
79
3"03
2
.0787
28
1. 1024
54
2.1260
80
3.1496
3
.1181
29
I.I417
55
2.1654
81
3.1890
, 4
1575
3<5
I.181I
56
2.2047
82
3.2284
5
.1968
31
1.2205
57
2.2441
83
3.2677
6
.2362
32
1.2598
58
2.2835
84
33071
7
.2756
33
1.2992
59
2.3228
85
33465
8
.3150
34
1.3386
60
2.3622
86
33859
9
•3543
35
1.3780
61
2.4016
87
34252
10
.3937
36
14173
62
2.4410
88
3.4646
II
.4331
37
1.4567
63
2.4803
89
35040
12
.4724
38
1.496 1
64
2.5197
90
35433
13
.5118
39
15354
65
2.5591
91
35827
14
.5512
40
1.5748
66
2.5984
92
3.6221
; '5
.5906
41
1. 6142
67
2.6378
93
3.6614
16
.6299
42
1.6536
68
2.6772
94
3.7008
17
.6693
43
1.6929
69
2.7166
95
37402
18
.7087
44
17323
70
2.7559
96
3.7796
19
.7480
45
1.7717
71
2.7953
97
3.8189
20
.7874
46
1.8110
72
2.8347
98
3.8583
1 21
.8268
47
1.8504
73
2.8740
99
38977
' 22
,8661
48
1.8898
•74
2.9134
100
39370
23
.9055
49
1. 9291
75
2.9528
= 1 de
cimetre.
24
.9449
50
1.9685
76
2.9922
' 25
.9S43
51
2.0079
77
30315
i ^'
1.0236
52
2.0473
78
3.0709
152
WEIGHTS AND MEASURES.
Tables No. 38 {continued),
InCHBS DbCIMALLY = MiLLIMBTRBS.
Inches.
1
Millimetres, i
Inches.
Millimetres.
Inches.
Millimetres.
Inches.
Millimetres.
239
.01
5 1
.26
6.60
.60
152
.94
.02
•51
.28
7.II
.62
157
.96
24.4
•03
.76
•30
7.62
.64
16.3
.98
249
.04
1.02
.32
8.13
.66
16.8
1. 00
25.4
•05
1.27
•34
8.64
.68
173
2.00
50.8
.06
152
.36
9.14
.70
17.8
3.00
76.2
.07
1.78
.38
9^65
.72
18.3
4.00
IOI.6
.08
2.03
.40
10.2
.74
18.8
5.00
127.0
.09
2.29
.42
10.7
.76
193
6.00
152.4
.10
2.54
.44
II. 2
.78
19.8
7.00
177.8
.12
3.05
.46
II.7
.80
20.3
8.00
203.2
.14
3.56
.48
12.2
.82
20.8
9.00
228.6
.16
4.06
•50
12.7
.84
21.3
10.00
254.0
.18
4.57
•52
13.2
.86
21.8
11.00
279.4
.20
5.08
•54
137
.88
22.4
12.00
304.8
.22
559
.56
14.2
.90
22.9
= I
foot.
.24
6.10
.58
14.7
•92
23.4
Inches
IN Fractions = Millimetrks.
Eighths.
Sixteenths.
Thirtyseconds.
Millimetres.
Eighths.
Sixteenths.
Thirtyseconds.
Millimetres.
I
•79
17
135
I
2
159
9
18
143
3
2.38
19
I5I
I
2
4
317
5
10
20
159
5
397
21
16.7
3
6
4.76
II
22
175
7
556
23
18.3
2
4
8
6.35
6
12
24
19.0
9'
7.14
25
19.8
5
10
7^94
13
26
20.6
II
8.73
27
21.4
3
6
12
9^52
7
14
28
22.2
13
10.32
29
23.0
7
14
II. II
15
30
23.8
15
II. 91
31
24.6
4
8
16
12.7 8
16
32
25.4
By means of the preceding tables of equivalent values of inches and
millimetres, the equivalent values of inches in centimetres and decimetres,
and even in metres, may be found by simply altering the position of the
decimal point. This method naturally follows from the decimal subdivisions
of French measure.
Take, for example, the tabular value of i millimetre, and shift the
IMPERIAL AND METRIC EQUIVALENTS.
153
decimal pK>int successively, by one digit, towards the righthand side; the
values of a centimetre, a decimetre, and a metre are thereby expressed in
inches, as follows: —
I millimetre 0394 inches.
I centimetre o394
I decimetre 3.94
I metre 39.4
At the same time, it appears that, by selecting the tabular value of
10 millimetres, the value of its multiples are given more accurately, thus, —
10 millimetres, or i centimetre 03937 inches.
I decimetre 3937 »»
I metre 39.37 „
Again: —
100 millimetres, or i decimeti^ = 3937 inches.
I metre =3937 n
Similarly, for example : —
.32 inch = 8.13 millimetres.
3.2 „ = 81.3 „
^ f 813.0 „ or
( .813 metre.
32.0
»
II. Square Measures, or Measures of Surface. — Tables No. 39.
Mbtric
I square centimetre
1 square decimetre
I square metre, or centiare
I ARE, or square dekametre, or 100
square metres
1 hectare, or metrical acre, or 100 ares,
or 10,000 square metres
Imperial Square Measures.
.155 square inch.
15.5003 square inches.
10.7641 square feet, or
1. 1960 square yards.
1076.41 square feet, or
119.60 square yards.
1,960.11 square yards, or
2.4711 acres, or
acres and 2280.1240 square
yards.
Imperial = Metric Square Measures.
I
Imperial Measures.
S<}uare
Centimetres.
Square Metres.
Ares.
Hectares.
f snuare incli r
= 6.45148
= 0.092901
= 0.836 II 2
= 25.292
= 1011.696
= 4046.782
= 10.11696
= 40.4678
= 0.40468
= 258.98944
1 square ft., or 144 sq. inches
I square yard, or 9 square )
tcet, or 1296 sq. inches )
I perch or rod, or 30X 
square yards y
I rood, or 40 perches, or")
I 1 2 10 square yards )
I acre, or 4 roods, or 4840 )
square yards J
I square mile, or 640 acres
154
WEIGHTS AND MEASURES.
III. Cubic Measures. — Tables No. 40.
Metric = Imperial Cubic Measures.
I cubic centimetre = 0.061025 cubic inch.
u* J ^ 4 (61.02524 cubic inches, or
I cubic deametre =  0.0353156 cubic foot
, . _, , f 35.^156 cubic feet, or
I cubic metre = { 1.308 cubic yanis.
Imperial = Metric Cubic Measures.
1 cubic inch = 16.387 cubic centimetres.
I cubic foot
I cubic yard
_ 128.31531 cubic decimetres, or
~ ( 0.02I
1283 1 6 cubic metre.
0.76453 cubic metre.
Wood Measure.
«'«. " »"« »"" { ^fisfcS^jSa
I decistfere * 3.5316 cubic feet.
I voie de bois (wood), or 2 stores. Paris { '^it'.l^^^y^^l
I voie de charbon (charcoal) = i sack ( S}^ bushels, or
= */5 stfere ( 7.063 cubic feet
I corde of wood = 4 cubic metres 141.26 cubic feet
IV. Measures of Capacity. — Tables No. 41.
or
Metric Dekominations
AND Values.
Equivalents in Imperial Denominations.
Litres.
Gills.
Pints.
Quarts.
Gallons.
Bushels.
a
Quarters.
Centilitre
Decilitre
Litre
(61.02524c. in.)
Dekalitre
Hectolitre
Kilolitre
/lOO
}'
10
100
1000
0.0704
0. 7043
0.0176
O.I761
1.7607
0.8804
0.2201
2.2009
22.009
220.09
0.2751
2.75II
27.511
0.344
3.439
Imperial Denominations.
Equivalents in Metric Denominations.
Litres.
Dekalitres.
Hectolitres.
I gill
I pint, or 4 gills
I quart, or 2 pints
I gallon, or 4 Quarts
= 0.1420
= 0.5679
= I.I359
= 4.5436
= 9.0872
= 36.3488
= 290.7904
= 0.9087
= 3.6349
= 29.0790
= 2.9079
I peck, or 2 crallons
I bushel, or 8 gallons
I Quarter, or 8 bushels
IMPERIAL AND METRIC EQUIVALENTS.
ISS
V. Measures of Weight. — Tables No. 42.
MsTKic Weights = Impbkial AvoiitDUPOis Weights.
I kilogramme = 2 lbs. 3 oz. 4 drachms^ 10.47374 grains.
Mbthic Weights.
Equivalbnts im Imperial Denominations.
Grammes.
Grains.
Ounces.
1
Pounds.
Hundred
weights.
Tons.
M illigramme
Vxooo
Vioo
I
10
100
1,000
10,000
100,000
1,000,000
0.01 54
0.1543
1.5432
154323
154.3235
1543.2349
15432.3487
0.3527
3.5274
35.2739
2.2046
22.0462
220.4621
2204.6212
1.9684
19.6841
1 II ! lit
CentigrazDzne
Decieramme
Gramme
Dekagramme
Hectogramme
Kilogramme
Myriagramme
Quintal^ or 100 kilog.
Millier, or metric ton
Imperial Avoirdupois = Metric Weights.
iMPBfiiAi. Avoirdupois
Weights.
Grammes.
Decigrammes.
Kilogrammes.
Millier, or
Metric Ton.
I diachni
= I.77184
 28.34954
= 453.59265
= 2.83495
= 45.35926
0.45359
== 50.80237
= IO16.O4754
= 1. 01604
I ounce, or 16 drams
I pound, or 1 6 ounces
I hundredweight, )
or 112 pounds )
I ton, or 20 hun \
dredweights j
Metric Weights = Imperial Troy Weights.
I kilogramme = 2 troy lbs. 8 oz. 3 dwts., .34874 grain.
Hetkxc Weights.
Grains.
Pennyweights.
Ounces.
Troy Pound.
1
Milligramme...
0.01543
^i
Centigramme ...
0.15432
Deagramnie . . .
1.54323
GRABHtfE.
1543234
Dekagramme...
= '154.32349
= 0.64301
= 0.32151
Hectogramme..
= 1543.23487
= 6.43014
= 321507
Kilogramme...
= I51432.34874
= 32.15073
= 2.67922
156
WEIGHTS AND MEASURES.
Imprxial Troy = Metric Weights.
Imperial Troy Weights.
Equivalents in Metric Denominations.
Millignuiime.
Gramme.
Dekagramme.
Hecto
gramme.
Kilo
gramme.
I troy grain
I „ dwt, or 24 gr.
I „ oz., or 480 „
I „ lb., or 5,760 „
64.79895
0.06480
155517
31.10349
37324195
3.IIO35
37.32419
373242
037324
APPROXIMATE EQUIVALENTS OP ENGLISH AND
FRENCH MEASURES.
The following are approximately equal English and French measures of
length : —
I pole, or perch {s}4 yards)... 5 metres (exactly 5.029 metres).
I chain (22 yards) 20 metres (exactly 20. 1 1 66 metres).
I furlong (220 yards) 200 metres (exactly 201.166 metres).
5 furlongs I kilometre (exactly i .0058 kilometres).
^ . (3 decimetres (exactly 3.048 decimetres), or
^ ^ \ 30 centimetres.
One metre = 3.28 feet = 3 feet 3 inches and 3 eighths all but Vsxa inch;
= 40 inches nearly ( ^/e^th. or 1.6 per cent less).
.100 metre (i decimetre)
.010 metre (i centimetre) , , __ , , ^.
.001 metre (i millimetre) = .04 inch, or Viooths inch, or twothirds
of Vx6 inch, or 725 inch, nearly.
One inch is about 2}4 centimetres (exactly 2.54).
One inch is about 25 millimetres (exactly 25.4).
One yard is "/"ths of a metre. 11 metres are equal to 12 yards.
Approximate rule for converting metres, or parts of metres, into yards : —
Add Vxi^h {}( per cent. less).
For converting metres into inches: — Multiply by 40; and to convert
inches into metres, or parts of metres, divide by 40.
One kilometre is about ^ mile (it is 0.6 per cent. less).
One mile is about 1.6 or 1 3/^ kilometres (it is 0.6 per cent less) == 16 10
metres, about.
«
With respect to superficial measures : —
One square centimetre is about 7*6.5 part of a square inch.
One square inch is equal to about 6.5 square centimetres.
One square metrecontains fully 10 J^ square feet, or nearly i^j square 3rards.
One square yard is nearly ^7 ths of a square metre.
One acre is over 4000 square metres (about 1.2 per cent more).
One square mile is nearly 260 hectares (about 0.4 per cent less).
FRENCH AND ENGLISH COMPOUND UNITS. 1 57
With respect to cubic measures, and to capacity : —
One cubic yard is about 5^ cubic metre (it is 2 per cent. more).
One cubic metre is nearly i^ cubic yard (it is i^ per cent. less).
One cubic metre is nearly 35 'A cubic feet (it is .05 per cent. less).
One litre is over i^ pints (it is 0.57 per cent more).
One gallon contains above 4^ litres (it holds about i per cent. more).
One kilolitre (a cubic metre) holds nearly i ton of water at 62*^ F. (i^
per cent less). — One cubic foot contams 28.3 litres.
With respect to weights: — The ton and the gramme stand at nearly
equal distances above and below the kilogramme, thus : —
I ton is 1,016,047.5 grammes,
I kilogramme is 1,000.0 grammes,
I gramme i.o gramme,
in the ratio of about 1,000,000 : 1,000 : i.
One gramme is nearly 15^ grains (about yi per cent. less).
One kilogramme is about 2 '/s pounds avoirdupois (about V4 per cent,
more).
A thousand kilogrammes, or a metric ton, is nearly one English ton
(about I J^ per cent less).
One hundredweight is nearly 5 1 kilogrammes ( 2/5 per cent less).
EQUIVALENTS OF FRENCH AND ENGLISH COMPOUND
UNITS OF MEASUREMENT.
Weighty Pressure^ and Measure,
• kiX*" P« "■""••■ { ;IM JSjl'^Taii
I pound per foot 1.488 kilogrammes per metre.
1 pound i>er yard .496 kilogramme per metre.
1000 kilc^ammes per metre .300 ton per foot
I ton per foot 3333333 kilogrammes per metre.
1000 kfl^mmes, or X tonne, per ) ^^g^ ^^^ ^^ ^^^
I tcm per mile 631.0 kilogrammes per kilometre.
, T .„../ 1422.32 pounds per square inch.
I kilogramme per square millimetre \ l^^\^^^ ^^^ ^^^^^^^ ^^^^
1000 Tiounds ner souare inch i 703077 kilogramme per square
1000 pounas per square men  millimetre.
, ton per square inch { '"575 ^"^JSTeSe.^" ^' .
I kilogramme per square centimetre 14.2232 pounds per square inch.
10335 kilogrammes per square centi ) nounds oer souare inch
metre (i atmosphere) / ^^'^ pounds per square men.
I pound per square inch
I pound per square foot
.0703077 kilogramme per square
centimetre.
4.883 kilogrammes per square
metre.
158
WEIGHTS AND MEASURES.
Weighty Pressure^ and Measure {continued),
kilogramme per square metre 205 pounds per square foot.
centimetre of mercury 394 inch of mercury.
inch of mercury 2. 540 centimetres of mercury.
centimetre of mercury 193 pound per square inch.
pound per square inch 5i7o centimetres of mercury.
gramme per litre 70.105 grains per gallon.
grain per gallon 0143 gramme per litre.
kilogramme per cubic metre 0624 pound per cubic foot
pound per cubic foot 16.020 kilogrammes per cubic metre.
. ^ u _ * f .984 ton per cubic metre.
tonne per cubic metre < \. *, u ^
^ ( .752 ton per cubic yard.
kilogramme per litre 10.016 pounds per gallon.
pound per gallon 998 kilogramme per litre.
ton per cubic metre 1.016 tonnes per cubic metre.
ton per cubic yard i'329 tonnes per cubic metre.
cubic metre per kilogramme 16.020 cubic feet per pound.
cubic foot per pound 06 24 cubic metre per kilogramme.
( 1.329 cubic yards per ton.
cubic metre per tonne < 1.794 cubic feet per cwt.
I 35.882 cubic feet per ton.
cubic yard per ton 752 cubic metre per tonne.
cubic foot per cwt. . . .^ 557 cubic metre per tonne.
cubic foot per ton 0279 cubic metre per tonne.
Volumty Area, and Length,
cubic metre per lineal metre 1.196 cubic yards per lineal yard.
cubic yard per lineal yard ..• 836 cubic metre per lineal metre.
cubic metre per square metre 3.281 cubic feet per square foot
cubic foot per square foot 3.048 cubic metres per square metre
litre per square metre 0204 gallon per square foot
gallon per square foot 48. 905 litres per square metre.
i .405 cubic metre per acre,
cubic metre per hectare \ .529 cubic yard per acre.
( 89.073 gallons per acre.
cubic metre per a^re 2.471 cubic metres per hectare.
cubic yard per acre 1.902 cubic metres per hectare.
000 gallons per acre 11.226 cubic metres per hectare.
Work,
kilogrammetre {ky.m) 7.233 footpounds.
footpound 138 kilogrammetre.
chevalvapeur or cheval (75 t x « ) horsepower.
persecond) / ^ ^ ^
horsepower 1.0139 chevaux.
kilogramme per cheval 2.235 pounds per horsepower.
pound per horsepower 447 kilogramme per cheval.
square metre per cheval 10.913 square feet per horsepower.
square foot per horsepower 0916 square metre per cheval.
cubic metre per cheval • 35.801 cubic feet per horsepower.
cubic foot per horsepower 0279 cubic metre per cheval.
FRENCH AND ENGLISH COMPOUND UNITS.
159
Heat.
I calorie, or French unit
I English heatunit
French mechanical equivalent (424 ) ,
kilogrammetresj exactly 423.55) ... j "^^ ^'
English mechanical equivalent (772 )
footpounds) J
I calorie per square metre
I heatunit per square foot
I calorie per kilogramme
1 heatunit per pound
10.
•
2.
I.
968 English heatunits.
252 calorie.
5 footpounds.
76 kilogrammetres.
369 heatunit per square foot.
713 calories per square metre.
800 heatunits per pound.
556 calorie per kilogramme.
Speedy 6fc.
I
I metre per second
I kilometre per hour
I foot per second, or per minute <
^ \ 1.609 kilometres per hour.
, ^.t _^. J f 35.316 cubic feet per second.
I cubic metre per second < ^^^ ^ x^ r ^ • *^
* ( 2119 cubic feet per mmute.
I cubic foot per second, or per minute
I cubic metre per minute
I cubic yard per minute
3.281 feet per second.
196.860 feet per minute.
2.236 miles per hour.
.621 mile per hour.
.305 metre per second,
minute.
.447 metre per second.
1.609 kilometres per hoi
or per
.02
Money,
I firanc per kilogramme
I penny per pound
I shilling per pound
I shilling per cwt., or £,1 per ton... <
I franc per quintal
I franc per tonne <
I fianc per metre \
1 shilling per yard
I franc per kilometre <
£1 per mile
I penny per mile
I franc per square metre \
:o cubic teet per secona.
cubic feet per minute.
583 cubic metre per second,
or per minute.
1.308 cubic yards per minute.
.765 cubic metre per minute.
4.320 pence per pound.
.360 shilling per pound.
40.320 shillings per cwt., or
^^40.32 per ton.
.231 fra^c per kilogramme.
2.772 franc per kilogramme.
24.802 francs per tonne.
2.48 francs per quintal.
.403 shilling per cwt.
.484 penny per cwt.
.806 shilling per ton.
.726 shilling per yard.
8.709 pence per yard.
1.378 francs per metre.
.0638 J[, per mile.
15.326 pence per mile.
15.660 francs per kilometre.
.0652 francs per kilometre.
8.028 pence per square yard.
.669 shilling per square yard
l6o WEIGHTS AND MEASURES.
I shilling per square yard 1.510 francs per square metre.
jCi per square yard 30. 194 francs per square metre.
{.270 penny per cubic foot
7.281 pence per cubic yard
.607 shilhng per cubic yard.
•0303 £ per cubic yard.
I penny per cubic foot 3. 708 francs per cubic metre.
I penny per cubic yard 137 franc per cubic metre.
I shilling per cubic yard i .648 francs per cubic metre.
j£i per cubic yard 32.962 francs per cubic metre.
I fianc per Utre { 43270 pence per gallon.
^ \ 3.606 shillmgs per gallon.
I franc per hectolitre 1893 shillings per hogshead (wine).
I shilling per hogshead 528 franc per hectolitre.
GERMAN EMPIRE. — ^WEIGHTS AND MEASURES. — Tables No. 43.
From the ist January, 1872, the French metric system of weights and
measures became compulsory throughout the German Empire, as follows : —
I. German Measures of Length.
French Measure.
I Strich = I millimetre,
10 Strichs I NewZoll = i centimetre.
100 NewZolls I Stab = i metre.
10 Stabs I Kette = i dekametre.
100 Kettes I Kilometre = i kilometre.
7 Kilometres i Mile  i 7°°^ metres, or
7 Kilometres i Mile   ^^^ ^^^gj^j^ ^y^^^
II. German Measures of Surface.
I QuadratStab = i square metre.
1 00 QuadratStabs i Ar = 1 00 square metres.
. Tj f _ / '^°^ square metres, or
'°^^^ I nectar ^i .247 acre.
={'■
III. German Measures of Capacity.
I Schoppen = j4 litre.
(Beer Measure.)
2 Schoppens i Kanne = i litre.
5° Cannes i Scheffel (bushel) = { ^^.Jj'l^^^erial bushels.
' S^J^effels X Fass (cask) = { J.Jf^^;
The kanne is further divided into measures of j^ kanne, ^ kanne, and
^/x6 kanne.
GERMAN EMPIRE. — WEIGHTS, THE FUSS. l6l
IV. German Measures of Weight.
I Milligramm = i milligramme.
lo Milligramms i Centigramm = i centigramme.
lo Centigramms i Dezigramm = i decigramme.
loo Dezigramms i NewLoth =  '"^ grammes, or
I 35273 ounce.
( 500 grammes, or
50 NewLoths........ i Pfund = < ^ kilogramme, or
( I . I o 2 3 pounds avoirdupois.
100 Pfunds I Centner = \ 5° kilogrammes, or
\ 110.23 pounds avoirdupois.
2 Centners, or ) Tonne = i ^°^ kilogrammes, or
200 Pfunds / '" ~ I 220.46 pounds avoirdupois.
OLD WEIGHTS AND MEASURES OF THE GERMAN STATES.
These vary for every state. The chief measures of length are the Fuss,
and the EUe, of which the second is in general twice the first. The
following are the values of the Fuss, which is the German foot, in the
principal states.
Values of the German Fuss in the States and Free Towns of
THE German Empire. — Table No. 44.
Prussia
Bavaria
Wiirtembeig
Saxony
Baden
MecklenburgSchwerin
HesseDarmstadt
HesseCassel
Oldenburg
Brunswick
Hanover
MecklenburgStrelitz
Anhalt
SaxeCoburgGotha
SaxeAltenbuig
Waldeck
Lippe
SchwarzburgRudolstadt
SchwarzburgSondershausen : —
(i) High Sovereignty and Amstadt ...
(2) Low Sovereignty and Sondershausen
Reuss
SchaumburgLippe
Hamburg
Liibeck
Bremen
2.356 inches.
1.491
1.279
1.149
1.811
I.4S7
9843
1.328
1.649
1.235
1.500
I.4S7
2.356
1.324
1. 122
1. 512
1.398
5047
1. 149
1.331
1.280
1. 421
1.283
1.324
1.392
11
i9
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
l62 WEIGHTS AND MEASURES.
KINGDOM OF PRUSSIA.— Old Weights and Measures.—
Tables No. 45.
I. Prussiai* Measures of Length.
English Measure.
I Linie = .0858 inch.
12 Linien i ZoU = 1.0297 inches.
12 ZoU I Fuss = j "356 inches, or
I 1.0297 feet.
2 Fuss I EUe = 2.0596 feet
1!?!?°'} iRuthe = 4.1192 yards.
R"* xMeae = { ^^^^Ja^^taS.
Used by Miners,
I Lachterlinie = .0927 inch.
I o Lachterlinien i Lach terzoU = .9268 inch.
10 Lachterzoll i Achtel = .7723 foot
8 Achtels, or I t i.*. a jx
5 p^gg I I Lachter = 2.0596 yards.
9 Fuss i Spanne =6.1788 yards.
Surveyor^ Measure,
I Scrupel = .0148 inch.
10 Scrupel I Linie = .1483 inch.
I o Linien i Zoll = 1.4828 inches.
10 ZoU I LandFuss = 1.2356 feet.
10 LandFuss i Ruthe =4.1192 yards.
2000 Ruthen i Meile = 4.6809 miles.
II. Prussian Measures of Surface.
I Square Linie = .00736 square inch.
144 Square Linien i Square Zoll  1.0603 square inches.
144 Square Zoll i Square Fuss = 1.0603 square feet
144 Square Fuss i Square Ruthe =16.967 square yards.
180 Square Ruthen... i Morgen = .63103 acre.
30 Morgen i Hufe = 18.931 acres.
III. Prussian Measures of Volume.
Cubic Measure,
I Cubic Linie = .000632 cubic inch.
1728 Cubic Linien.... i Cubic Zoll = 1.092 cubic inches.
1728 Cubic Zoll I Cubic Fuss = 1.092 cubic feet
1728 Cubic Fuss I Cubic Ruthe =69.893 cubic yards.
For measuring stone and brickwork, earth, peat, fascines, and firewood,
the following are u§e4 ;—
PRUSSIA. — CAPACITY, WEIGHTS. 1 63
I Cubic Klafter, or ) . , . .
. 108 Cubic Fuss ;  " 7.93 cubic feet
4ji Klafters i Haufe =530.70
I Schachnithe (in architecture) 144 Cubic Fuss = 157.25
99
IV. Prussian Measures of Capacity.
Dry Measure,
I Maasche = •7560 quart.
JSS'"'"} ^""^^ = 34. quarts.
4 Metzen .. i Viertel = 3.0242 gallons.
4 Viertel, or ) ^ Scheffel = / ^'S'^' bushels, or
48 Quarts ) ( i. 941 cubic feet.
4 Scheffeln i Tonne = 6.0484 bushels.
if 2S?r } ' Maker = 2..68rs quarters.
6jSeSn°'} '^' = I X.3407 quartets.
The Tonne in the table is the measure for salt, lime, and charcoal.
A Tonne of flaxseed is 2.354 Scheffeln.
Liquid Measure (for Wine and Spirits).
32 Cubic ZoU I Ossel = 1.0079 pints.
2 Ossel I Quart = 1.0079 quarts.
i:§Slf'" } ^^''^ = 7.559 gallons.
2 Ankers i Eimer = 15. 118
2 Elmers i Ohm = 30.237
ffS°' } ^Oxhoft =45.355
V. Prussian Measures of Weight.
I Com = 4. 1 15 grains.
10 Corns I Cent = .09406 dram.
10 Cents I Quentche = .9406 dram.
10 Quentchen i Loth = ,588 ounce.
30 Loth I Zollpfund = 1. 1023 pounds.
1 00 Zollpfund I Centner = 110.23 pounds.
20 Zollpfund I Stein = 22.046 pounds.
330.69 pounds, or
2.506 hundredweights.
4409.2 pounds, or
3 Centners i Schiffspfund = ^
40 Centners I x Schiffslast = { '^'^"^g^g™;
The Tonne of coals is 2270 pounds avoirdupois, or 1.013 tons.
1 64 WEIGHTS AND MEASURES.
KINGDOM OF BAVARIA.— Old Weights and Measures.—
Tables No. 46. *
I. Bavarian Measures of Length.
I Linie = .6798 inch.
12 Linien i ZoU = .95756 inch.
12 Zoll I Fuss = .95756 foot
6 Fuss i Klafter=5.74536 feet.
10 Fuss i Ruthe =9.5756 feet
In surveying, the Fuss is divided into 10 Zoll, and i Zoll into 10 Linien.
The EUe contains 2 Fuss lo^ Zoll, = 2.733 ^^^^
II. Bavarian Measures of Surface.
I Square Zoll = .91692 square inch.
144 Square Zoll .... i Square Fuss = .91692 square foot
100 Square Fuss ... i Square Ruthe = 10.188 square yards.
400 Square Ruthen / ' Tagwerk Morgen, ) ^ ( 4075.188 square yards, or
^ I or Juchert J  .842 acre.
III. Bavarian Measures of Volume.
I Cubic Zoll = .878 cubic inch.
1728 Cubic Zoll I Cubic Fuss= .878 cubic foot
126 Cubic Fuss (6x6 xyA Fuss) i Klafter = / "^'•^28 cubic feet or
' ( 4.097 cubic yards.
IV. Bavarian Measures of Capacity.
Dry Measure,
I Dreisiger= .12745 peck.
4Dreisigers i Maassl = .12745 bushel.
4 Maassls i Viertel = .5098 bushel.
2 Viertel i Metze =1.0196 bushels.
6 Metzen i Schaffel =6.1176 bushels.
4 SchafTel i Muth =3.0588 quarters.
Liquid Measure,
I Maaskanne= .23529 gallon.
64 Maaskannen i Eimer = 15.05856 gallons.
25 Eimer i Fass =376.464 gallons.
The SchenkEimer, ordinarily used in die Wine trade, contains only
60 Maaskannen, equal to 14. 1 174 imperial gallons.
V. Bavarian Measures of Weight.
I Quentchen= .15433 ounce.
4 Quentchen i Loth = .6173 ounce.
32 Loth I Pfund = 1.23457 pounds.
100 Pfund I Centner = [ ^^3457 pounds, or
( r.io2 hundredweights.
WiJRTEMBEKG.— LENGTH, SURFACE, ETC. 165
KINGDOM OF WURTEMBERG.— Old Weights and Measures.—
Tables No. 47.
I. WiJRTEMBERG MEASURES OF LENGTH.
I Punkte = .01128 inch.
10 Punkte I Linie = .1128 inch.
10 Linien i ZoU = 1. 128 inches.
10 Zoll I Fuss = .93995 foot
10 Fuss I Ruthe = 9.3995 feet.
2.144 Fuss I Elle = 2.015 feet
6 Fuss I Klafter = 5.6397 feet
26,000 Fuss I MeUe = i 8146.25 yards, or
' ( 4.6285 miles.
II. WiJRTEMBERG MEASURES OF SURFACE.
1 Square Zoll = 1.272 square inches.
100 Square ZolL i Square Fuss = 8835 square foot
100 Square Fuss i Square Ruthe = 88.3506 square feet
384 Square Ruthen... i Moigen = { 3769626 square yards, or
III. WiJRTEMBERG MEASURES OF VOLUME.
I Cubic Linie = .001434 cubic inch.
1000 Cubic Ltnien i Cubic Zoll = 1.434 cubic inches.
1 000 Cubic Zoll I Cubic Fuss = . 83045 cubic foot
144 Cubic Fuss I Cubic Klafter =119.583 cubic feet
IV. WiJRTEMBERG MEASURES OF CAPACITY.
I
Dry Measure,
I Viertlein = .305 pint
4 Viertlein i Ecklein =1.219 pints.
8 Ecklein i Vierling  1.2 19 gallons.
4 Vierling i Simri =4.876 gallons.
8 Simri i Scheffel = 4.876 bushels.
Liquid Measure,
I Quart or Schoppen = .4043 quart.
4 Quarts i Helleich Maass = 1.6173 quarts.
I o Helleich Maass i Irai = 40433 gallons.
16 Imi I Eimer = 64.6928 gallons.
6 Eimer i Fuder =388.1568 gallons.
V. WiJRTEMBERG MEASURES OF WEIGHT.
I Quentchen = .1289 ounce.
4 Quentchen i Loth . = .5156 ounce.
32 Loth I Light Pfund = 1.03115 pounds.
\V, S?.7huS*.."} ^ C»^ = ^^396 pounds.
100 Light Pfund = 103. 115 pounds.
1 66 WEIGHTS AND MEASURES.
KINGDOM OF SAXONY.— Old Weights and Measures.—
Tables No. 48.
I. Saxon Measures op Length.
I Linie = .07742 inch. •
2 Linien i ZoU = .9291 inch.
12 Zoll I Fuss = .9291 foot
2 Fuss I EUe = 1.8582 feet
2 Ellen I Stab = 3.7165 feet
15 Fuss, 2 Zoll I Ruthe (Land Measure^ = 4.6972 yards.
16 Fuss I Ruthe (Road Measure) = 4.9553 yards.
I Lachter (Mining) = 2.1873 yards.
1324.987 Ellen I Meile Post = 4.6604 miles.
II. Saxon Measures of Surface.
I Square Zoll = .8632 square. inch.
144 Square Zoll i Square Fuss = .8632 square foot
300 Square Ruthen i Acker = i . 4865 acres.
III. Saxon Measures of Volume.
I Cubic Zoll = .8021 cubic inch.
1728 Cubic Zoll I Cubic Fuss = .8021 cubic foot
108 Cubic Fuss I Klafter = 86.624 cubic feet
3 Klafter i Schragen =259.873 cubic feet
The. Klafter is 6 Fuss by 6 Fuss by 3 Fuss. The Schragen is used in
the measurement of firewood.
IV. Saxon Measures of Capacity.
Dry Measure,
I Maasche = 1.4463 quarts.
4 Maaschen i Metze = 1.4463 gallons.
4 Metzen i Viertel = 5.7852 gallons.
4 Viertel i SchefTel = 2.8926 bushels.
12 SchefFel i Malter =34.7124 bushels.
2 Malter i Wispel = 69.4249 bushels.
Liquid Measure,
I Quartier = .2059 pint
4 Quartier i Nossel = .8237 pint
2 Nossel I Kanne = 1.6474 pints.
36 Kannen i Anker = 7.4237 gallons.
2 Anker i Eimer = 14.8262 gallons.
3 Eimer i Oxhoft = 44.4687 gallons.
6 Eimer i Fass or Barrel = 88.9374 gallons.
V. Saxon Measures of Weight.
The old Saxon measures of weight are the same as those of Prussia.
BADEN. — LENGTH, SURFACE, ETC 1 67
GRAND DUCHY OF BADEN.— Old Weights and Measures.—
Tables No. 49.
I. Baden Measures of Length.
I Punkte = .0118 inch.
10 Punkte I Linie = .iiSi^jich.
10 Linien i ZoU = 1. 181 inches.
10 ZoU I Fuss = .9842 foot
2 Fuss I EUe = 1.9685 feet
10 Fuss I Ruthe = 9.8427 feet
6 Fuss I Klafter = 5.9055 feet
1 48 1 4.8 1 5 Fuss I Stunde =4860.59 yards.
2 Stunden i Meile = 5.5234 miles.
II. Baden Measures of Surface.
I Square ZoU = i395i square inches.
100 Square ZoU i Square Fuss = .9688 square foot
100 Square Fuss i Square Ruthe = 10.7643 square yards.
100 Square Ruthen... i Viertel = 1076.43 square yards.
4 ^itel X Morgan = { ^^^l^^S:/"'^' "
III. Baden Measures of Volume.
I Cubic Fuss = .95335 cubic foot
X 44 Cubic Fuss i Klafter = 137.28 cubic feet
IV. Baden Measures of Capacity.
Liquid Measure.
I Glass  1.0563 gills.
10 Glass I Maass = 1.3204 quarts.
10 Maass .*. i Stutze = 3.3014 gallons.
10 Stutzen I Ohm = 33.014 gallons.
10 Ohm I Fuder = 330.14 gallons.
Dry Measure.
I Becher = .2643 pint
10 Becher i Maasslein = .1652 peck.
10 Maasslein i Sester = .4127 bushel.
10 Sester i Maker = 4.1268 bushels.
10 Malter i Zuber = 41.2679 bushels.
V. Baden Measures of Weight.
I As = .7716 grain.
10 As I Pfennig = 7.716 grains.
10 Pfennig...., i Centas = .1764 ounce.
10 Centas i Zehnling = 1.7637 ounces.
10 Zehnling i Pfund = 1.1023 pounds.
100 Pfund iCentner = 110.230 pounds.
1 68 WEIGHTS AND MEASURES.
THE HANSE TOWNS. — OlD WEIGHTS AND MEASURES. —
Tables No. 50.
HAMBURG. — ^Weights and Measures.
I. Hamburg Measures of Length.
I Acht'el = .1175 inch.
8 Achtel I ZoU = .9402 inch.
12 ZoU I Fuss = . .9402 foot.
2 Fuss i Elle = 1.8804 feet
6 Fuss i Klafter, or Faden= 5.6413 feet
14 Fuss i MarschRuthe = 13.1629 feet
16 Fuss I GeestRuthe = 15.0434 feet
The Hamburg Elle above is used for silk, linen, and cotton goods. The
Brabant Elle is equal to i V5 Hamburg Elle; and 4 of them are reckoned
equal to 3 yards. The Prussian Ruthe is also used. The Prussian Fuss is
used in surveying.
II. Hamburg Measures of Surface.
I Square ZoU = .8840 square inch,
144 Square ZolL... i Square Fuss = .8840 square foot
196 Square Fuss... i Square MarschRuthe = 173.26 square feet
256 Square Fuss... i Square GeestRuthe = 226.30 square feet
200 Square Geest ) o^^effel Oe^t T^nd  i 5028.98 square yards, or
Ruthen / ^ ^^^^^^ oeestl^na  ^ ^^^^ ^^^^^
600 Sq. Marsch ) ^ ^ ^ f ii55o93 square yards, or
Ruthen... J ^ ( 2.386 acres.
III. Hamburg Measures of Volume.
I Cubic Zoll = .8311 cubic inch.
1728 Cubic 2k)ll I Cubic Fuss = .8311 cubic foot
88.9 Cubic Fuss.... I (Cubic) Klafter = 73.88 cubic feet
1 2 o Cubic Fuss i Tehr =9973 cubic feet
IV. Hamburg Measures of Capacity.
Liquid Measure.
I Ossel = .09965 gallon.
2 Ossel I Quartier = .1993 gallon.
2 Quartier i Kanne = .3987 gallon.
2 Kannen i Stubchen =• .7974 gallon.
I Stubchen i Viertel = 1.5947 gallons.
4 Viertel i Eimer ^ 6. 3 7 88 gallons.
5 Viertel i Anker = 7.9735 gallons.
6 Eimer i Tonne = 38.2 728 gallons.
4 Anker i Ohm = 31.8940 gallons.
6 Anker i Oxhoft = 47.8410 gallons.
6 Ohm I Fuder, or Tonneau= 191.3640 gallons.
The above are measures for Wines and Spirits. For Beer^ there are
three sizes of Tonne, containing respectively 48, 40, and 32 Stubchen.
HAMBURG. — WEIGHTS. 169
Dry Measure.
I Small Maass = .0236 bushel
2 Small Maass i Large Maass = .0473 bushel.
4 Laige Maass i Spint = .1890 bushel.
4 Spint I Himten = .7560 bushel.
2 Himten i Fass = 1.5121 bushels.
2 Fass I Scheffel = 3.0242 bushels.
10 Scheffeln i Wispel • = 30.2416 bushels.
3 Wispel I Last =90.7248 bushels.
For barley and oats, the Scheffel contains 3 Fass.
V. Hamburg Measures of Weight.
I Half Gramme = .0011 pound =.5 gramme.
10 Half Grammen i Quint = .01102 pound =5 grammes.
10 Quinten i (New) Unze = .11023 pound =50 „
10 (New) Unzen.. i (New) Pfund = i.io232pounds = 5oo „
100 (New) Pfund i Centner = 110.232 pounds = 50 kilog.
60 Centners. i (Commercial) Last = { ^or^^ q^.? ^^^ =3000 kilog.
This, it is apparent, is a metric system of weights, which was comparatively
recently introduced and adopted at Hamburg. It is now, of course, over
ruled by the French metric system enforced for the German Empire.
BREMEN. — Old Weights and Measures.
The Fuss is equal to 11.392 inches, and the Klafter is equal to 5.696
feet. The Morgen = .6368 acre. The principal measures for wines and
spirits are the Viertel =1.56 gallons; the Anker = 5 Viertels = 7.80 gallons;
the Oxhoft = 46.80 gallons. The Scheffel, for dry goods = 2.0388 bushels.
The old weights are the same as those of Hamburg.
LUBEC. — Old Weights and Measures.
The Fuss is equal to 11.324 inches. The Viertel =1.60 gallons; the
Anker = 8 gallons ; the Oxhoft = 48.04 gallons. The Scheffel, for dry goods,
= .9545 bushel. The old Pfund =1.0725 pounds, and the Centner =
1.0725 cwts.
GERMAN CUSTOMS UNION.— Old Weights and Measures.—
Table No. 51.
Centner 110.23 pounds (50 kilogrammes).
ShipLast of timber about 80 cubic feet.
Scheffel 1.512 bushels.
Klafter 6 feet.
In Oldenbuig, Hanover, Brunswick, SaxeAltenbourg, Birkenfeld, Anhalt,
Waldeck, Reuss, and SchaumburgI^ippe, the old system of weights is the
same as that of Prussia.
I/O WEIGHTS AND MEASURES.
AUSTRIAN EMPIRE. — WEIGHTS AND MEASURES. — ^Tables No. 52.
I. Austrian Measures of Length.
I Punkte = .0072 inch.
12 Punkte I Linie = .0864 inch.
12 Linien i 2k)ll = 1.0371 inches.
12 Zoll I Fuss = 1.037 1 feet
2 Fuss i EUe = 2.0742 feet
6 Fuss I Klafter = 6.2226 feet
4000 Klafter i Meile(post) = { ^^^J. J^^^.
II. Austrian Measures of Surface.
I Square Zoll = 1,0756 squareinches.
144 Square Zoll i Square Fuss = 1.0756 square feet.
36 Square Fuss i Square Klafter =  38.7"5 square feet, or
^ ^ ^uiu^ A ««Li.j ^1 4.3025 square yards.
8 }4 Square Klafter, or ) « d ^i.  o j
300 Square ^uss } i Square Ruthe = 35.854 square yards.
64 Square Ruthen i Metze = 2294.7 square yards.
3 Metzen, or ) * , _ f 6884 square yards, or
1600 Square Klafter J ^ " ( 1.4223 acres.
III. Austrian Measures of Volume.
Cudic Miosure,
I Cubic Zoll = I. "55 cubic inches,
1728 Cubic Zoll I Cubic Fuss = iii55 cubic feet
216 Cubic Fuss.... i Cubic Klafter = [ "4^94 cubic feet or
\ 8.924 cubic yards.
IV. Austrian Measures of Capacity.
Dry Measure.
iProbmetzen = ( '""SpinljOr
( 3.665 cubic inches.
8 Probmetzen i Becher = .8460 pint
4 Becher i Futtermassel = 1.6920 quarts.
2 Futtermassel i Muhlmassel = [ ^'tf ^T ^^' ^^
\ .8460 gallon.
2 Muhlmassel i Achtel = 1.6920 gallons.
2 Achtel I Viertel = [ 3384o gallons, or
( .4230 bushel.
4 Viertel i Metze = 1.69 18 bushels.
30 Meuen i Muth =/ So. 7536 bushels, or
^ \ 8.3442 quarters.
AUSTRIAN EMPIRE. — CAPACITY, WEIGHTS.
171
Liquid Measure.
I Pfiflf
2Pfiflf iSeidel =
2 Seidel i Kanne =
2 Kannen i Mass =
10 Mass I Viertel =
4 Viertel i Eimer =
32 Eimer.... i Fuder =
1.246 gills, or
10.781 cubic inches.
2.491 cubic inches, or
.6229 pint
1.2457 pints.
1.2457 quarts.
3. 1 143 gallons.
12.4572 gallons.
398.6304 gallons.
V. Austrian Measures of Weight.
{270.1 grains, or
.6173 dram.
2.4694 drams.
9.8776 drams, or
.6173 ounce.
1.2347 ounces.
4.9388 ounces.
9.8776 ounces, or
.6173 pound avoirdupois.
= 1.2347 pounds avoirdupois.
_ (123.47 pounds avoirdupois, or
~ ( 1. 1024 hundredweights.
In 1853, a pfimd of 500 grammes, with decimal subdivisions, was adopted
tor customs and fiscal purposes.
I Pfenning
4 Pfenning i Quentchen
4 Quentchen... i Loth
2 Loth I Unze
4 Unzen i Vierdinge
2 Vierdinges... i Mark
I.. I Pfund
100 Pfimd I Centner
2 Marks, or
16 Unzen
RUSSIA. — ^Weights and Measures. — ^Tables No. 53.
L Russian Measures of Length.
English Equivalent.
I Vershok = 1.75 inches.
16 Veishoks i Arschine = 28 „
3 Arschines i Sajene = 7 feet.
( 3500 feet, or
500 Sajenes i Verst = < ii66^ yards, or
( 0.6629 mile.
The Fuss, or Russian foot, is 13.75 inches; but, since 1831, the English
foot of 1 2 inches has been used as the ordinary standard of length, each
inch being divided into 12 parts.
I Lithuanian Meile 55574 English miles.
I Rhein Fuss, used in calculating ) t? r 1. r *
export duties on timber
/
1^2 WEIGHTS AND MEASURES.
II. Russian Measures of Surface.
o A I, .. f 784 square inches, or
I Square Arschine =  ^ J ^ ^^^ ^^
9 Square Arschines.. i Square Sajene =  ^^.^J^u^S'y^s.
2400 Square Sajenes i Desatine \ ^' ; ^ o^^r^c '
III. Russian Measures of Capacity.
Liquid Measure,
I o Tscharkeys i Kruschka = 1.0820 quarts.
1 00 Tscharkeys i Vedro = 2. 7049 gallons.
3 Vedros i Anker = 8.1 147 „
tz v! AnkeS } •••• ' Sarokowaja Boshka = 108.196
Dry Measure (Grain).
I Gamietz = 2.885 quarts.
2 Gamietz i Tschetwerka = i .4424 gallons.
4 Tschetwerkas... i Tschetwerik = .7213 bushel.
2 Tschetweriks.... i Pajak = 1.4426 bushels.
2 Pajaks I Osmin = 2.8852 „
2 Osmins i Tschetwert* = 5.7704 „
■« The*e«s . L». = { ".ijfi^JSKL
* A Tschetv^ert is usually reckoned as 5ji^ bushels, and 100 Tschetwerts as 72 quarters^
though they are more exactly 72.1308 quarters.
100 quarters are equal to 138.637 Tschetwerts.
For earthworks, masonry, &c., the Sajene is divided into tenths (dessiatka),
hundredths (sotka), and thousandths (tisiatchka), which are used as a basis
for lineal, superficial, and cubic measurements, similarly to the French
metre with its submultiples.
IV. Russian Measures of Weight.
96 Dolls I Zolotnick = <
I Dolis = .68576 grain.
65.833 grains, or
.1505 ounce.
3 2k)lotnicks... i Lotti = .4514 »
8 2k)lotnicks... i Lana = 1.2037 ounces.
12 Lanas, or \ i .90285 pound avoirdupois, or
32 Lottis, or > I Funt, or pound = < 14.446 ounces, or
96 2k)lotnicks j (6320 grains.
40 pounds I Pood = 36.114 pounds avoirdupois.
10 Poods I Berkovite = \ 36^^4 pounds avoiidupois. or
( 3.224 hundredweights.
3 Berkovitz i Packen = 9.672 hundredweights.
HOLLAND, BELGIUM, NORWAY, ETC. I73
62.0257 Poods I English ton.
2481.0268 Russian pounds i „
The Pood is commonly estimated at 36 pounds avoirdupois.
The Nurembeig pound, used for apothecaries' weight, weighs 5527 grams,
or about .96 pound troy.
The ShipLast is equal to 2 tons.
The Caraty for weighing pearls and precious stones, is about 3 ^6 grains.
HOLLAND.
The metric system was adopted in Holland in 1819; the denominations
corresponding to the French are as follows : —
Ltngth, — Millimetre, Streep; centimetre, Duim; decimetre, Palm; metre,
El; decametre, Roede; kilometre, Mijle.
Surface, — Square millimetre, Vierkante Streep; square centimetre, Vier
kante Duim; and so on. Hectare, Vierkante Bunder.
Cubic MecLsure, — Millistere, Kubicke Streep, and so on.
Capacity, — Centihtre, Vingerhoed; decilitre, Maatje; liquid litre, Kan;
dry litre, Kop; decalitre, Schepel; Hquid hectolitre. Vat or Ton; dry
hectolitre. Mud or Zak; 30 hectolitres = i Last= 10.323 quarters.
Weight, — Decigramme, Korrel; gramme, Wigteje; decagramme, Lood;
hectogramme, Onze; kilogramme, Pond.
BELGIUM.
The Frenc3i metric S)rstem is used in Belgium. The name Livre is
sabstitated for kilogramme, Litron for litre, and Aune for metre.
NORWAY AND DENMARK.
Weights and Measures. — Tables No. 54.
I. Norwegian and Danish Measures of Length.
I Linie = .0858 inch.
12 Linier i Tomme = 1.0297 inches.
12 Tommer i Fod = 1.0297 feet
2 Fod I Alen = 2.0594 „
3 Alen, or
6 Fod
2 Favn, or
12 Fod
2,000 Roder, or ) j^^ ^ i 823777 yards, or
> I Favn = 6.1783 „
I Rode = 12.3567 „
24,000 Fod
( 4.68055 miles.
23,642 Fod I nautical mile= 4.61072 English miles.
IL Norwegian and Danish Measures of Surface.
144 Square Linie i Square Tomme = 1.0603 square inches.
144 Square Tomme... i Square Fod = 1.0603 square feet
144 Square Fod i Square Rode = 16.966 square yards.
174 WEIGHTS AND MEASURES.
III. Norwegian and Danish Measures of Volume.
1728 Cubic Linier i Cubic Tomme = 1.0918 cubic inches.
1728 Cubic Tomme.... i Cubic Fod = 1.0918 cubic feet
The Favn of firewood measures 6x6x2 Fod= 72 cubic Fod = 78.60
cubic feet In forest measure it is 6)4 x 6j4 x 2 Fod = 84)^ cubic Fod =
92.26 cubic feet
IV. Norwegian and Danish Measures of Capacity.
Liquid Measure,
I Paegle = .4248 pint
4 Paegle i Pot = 1.6991 pints.
2 Potter I Kande = 3.3983 „
38 Potter I Anker = 8.0709 gallons.
136 Potter I Tonde = 28.885 * >>
6 Ankeme i Oxehoved = 48.4256 „
4 Oxehoveder i Fad =193.7027 „
Dry Measure.
I Pot = 1.699 1 pints.
18 Potter I Skeppe = 3.8232 gallons.
2 Skepper i Fjerdingkar = .9558 bushel.
4 Fjerdingkar i Tonde = 3.8231 bushels.
12 Tender i Laest =AS'^T^9 »
V. Norwegian and Danish Measures of Weight.
I Ort = 7.7163 grains.
10 Ort I Kvint = 77.163 „
100 Kvinten i Pund = 1.1023 pounds.
100 Pund I Centner =110.23 „
40 Centner i Last = i . 9684 tons.
52 Centner i SkipLast = 2.5590 „
16 Pund I Lispund = 17.63 7' pounds.
320 Pund I Skippund = 3.149 cwts.
SWEDEN. — Weights and Measures. — Tables No. 55.
I. Swedish Measures of Length.
I Linie = .1169 inch.
10 Linier i Turn = 1. 1689 inches.
10 Tumer i Fot = 11.6892 „
10 Fot I Stang = .9.7411 feet
10 Stanger i Ref = 32.4703 yards.
760 Ref I Meile  / ">6S93o8 yards, or
^^"^ ^^* ' ^"^^  \ 6.6417 miles.
2 Fpt I Aln = 1942 feet
6 Fot I Faden =^ 5.845 „
SWEDEN, SWITZERLAND. 1 75
II. Swedish Measures of Surface.
loo Square Linier... i Square Turn = 1.3666 square inches.
100 Square Turner., i Square Fot = 9489 square foot
1 00 Square Fot i Square Stang = 3* 5 1 46 square yards.
100 Square Stanger i Square Ref = I •°54^ square yards, or
4 Square Fot i Square Aln = 37956 square feet.
5.6 Square Ref:... r Tunnlar^d = { "^Lp^^S"'" "
III. Swedish Measures of Volume.
Cttbic Measure,
I Cubic Turn =1.5972 cubic inches.
1000 Cubic Turner i Cubic Fot = .9263 cubic foot.
8 Cubic Fot I Cubic Aln = 7.4104 cubic feet
Liquid and Dry Measure.
1 000 Cubic Linier i Cubic Turn = . 1 843 gill.
1 00 Cubic Turner i Kanna = 2. 3096 quarts.
10 Kanna i Cubic Fot = 5.774 gallons.
8 Cubic Fot I Cubic Aln =46.192 „
IV. Swedish Measures of Weight.
I Kom = .6564 grain.
100 Kom I Ort = 2.4005 drams.
100 Ort I Skalpund = .9377 pound.
o Skalpur^d X Cer^tner = { 937739 pounds, or
IOC Centner i NyLast = 4.1892 tons.
A Pund, commercial, is .9377 pound.
A Pund, freight, is .75016 pound.
A Pund, miners* mark weight, is .8285 pound.
A Pund, country town's mark weight, is .7891 pound.
S^VITZERLAND. — ^WEIGHTS AND MEASURES. — Tables No. 56.
I. Swiss Measures of Length.
I Striche = .01181 inch.
10 Striche I Linie = .11811 „
10 Linien i 2k)ll = 1.181 12 inches.
10 Zoll I Fuss = 11.81124 „
2 Fuss i Elle = 1.9685 feet
6 Fuss i Klafter = 59056 ^
10 Fuss i Ruthe = 98427 „
1600 Ruthen .... i Schweizerstunde, or Lien= < 2 o82^^iles
176 WEIGHTS AND MEASURES.
II. Swiss Measures of Surface.
I Square ZoU = 13947 square inches.
100 Square ZoU i Square Fuss = .9688 square foot
36 Square Fuss i Square Klafter = 34.8768 square feet
100 Square Fuss i Square Ruthe = 10.7643 square yards
400 Square Ruthen. . i Juchart = .8694 acre.
6400 Jucharten i Square Stunde = 5693.52 acres.
350 Square Ruthen i Juchart, of meadow land.
450 Square Ruthen i Juchart, of woodland.
III. Swiss Measures of Volume.
I Cubic Zoll = 1.6476 cubic inches.
1000 Cubic ZolL I Cubic Fuss = .9535 cubic foot
216 Cubic Fuss I Cubic Klafter = 7.6172 cubic yards.
1000 Cubic Fuss I Cubic Ruthe =35.3166 „
IV. Swiss Measures of Capacity.
Dry Measure,
I Imi = 1.3206 quarts.
10 Imi I Maass = .4127 bushel.
10 Maass i Malter = 4.1268 bushels.
Liquid Measure,
2 Halbschoppen i Schoppen = 2.6412 gills.
2 Schoppen i Halbmaass = 1.3206 pints.
2 Halbmaass i Maass = 2.6412 „
100 Maass i Saum = 33.015 gallons.
V. Swiss Measures of Weight.
I Quntii = 2.2048 drams.
4 Quntli I Loth = .5511 ounce.
2 Loth I Unze = 1. 1023 ounces.
16 Unzen i Pfund = 1. 1023 pounds.
100 Pfund I Centner = 110.233 pounds, or .9842 cwt.
The Pfund is divided into halves, quarters, and eighths. It is also
divided into ^00 Grammes, and decimally into Decigrammes, Centi*
grammes, and Milligrammes.
SPAIN. — ^Weights and Measures. — Tables No. 57.
The French metric system was established in Spain in 1859. The metre
is named the Metro; the litre, Litro; the gramme, Grammo; the are, Area;
the tonne, Tonelada. The metric system is established likewise in the
Spanish colonies. The old weights and measures are still largely used.
SPAIN — LENGTH, SURFACE, ETC. 1 77
L Old Spanish Measures of Length.
I Punto = .00644 inch.
12 Puntos I Linea = .07725 inch.
12 Lineas i Pulgada = .927 inch.
6 Pulgadas i Sesma = 5.564 inches.
2 Sesmas i Pies de Burgos = .92 73 foot.
3 Pies de Burgos i Vara = 2.782 feet
2 Varas i Estado = 5.564 feet
4 Varas i Estadal =11.128 feet
5000 Varas i Legua (Castilian) = 2.6345 miles.
8000 Varas i Legua (Spanish) = 4.2 151 miles.
II. Old Spanish Measures of Surface.
I Square Pies = .860 square foot
9 Square Pies i Square Vara = .860 square yard.
16 Square Varas i Square Estadal = 13.759 square yards.
50 Square Varas .... i Estajo =42.997 square yards.
576 Square Estadals. i Fanegada = 1.6374 acres.
50 Fanegadas. i Yugada = 81.870 acres.
III. Old Spanish Measures of Capacity.
Liquid Measure.
I Capo = .888 gill.
4 Capos I Cuartillo = .111 gallon.
4 Cuartillos i Azumbre = .444 gallon.
2 Azumbres i Cuartilla = .888 gallon.
^ ^.« ( I ArrobaMayor,orCantara) ^ ^^ „«n^««
4 CuartUlas...  (for wine) j " 3552 gallons.
16 Cantaras i Mayo =56.832 gallons.
The old measure for oil is the Arroba Menor= 2.7652 gallons.
JDry Measure,
I Ochavillo = .00785 peck.
4 Ochavillos i Racion = .0314 peck.
4 Raciones i Quartillo = . 03 1 4 bushel.
2 Quartillos i Medio = .0628 bushel.
2 Medios i Almude = .1256 bushel;
12 Amuerzas i Fanega = 1.5077 bushels.
1 2 Fanegas i Cahiz = 1 8.0920 bushels.
IV. Old Spanish Weights.
I Grano = .771 grain.
12 Granos i Tomin = 9.247 grains.
3 Tomines .... I Adarme = 27.74 grains.
2 Adarmes .... i Ochavo, or Drachma = .1268 ounce.
8 Ochavos i Onza = i.o 144 ounces.
8 Onzas i Marco = 8.1 154 ounces.
2 Marcos i Libra (Castilia a) = i . o 1 44 pounds.
IOC Libras i Quintal = 101.442 pounds.
10 Quintals i Tonelada =1014.42 pounds.
19
178 WEIGHTS AND MEASURES.
PORTUGAL.
The French metric system of weights and measures v^as adopted in its
entirety during the years 186063, and was made compulsory from the ist
October, 1868, The chief old measures still in use are, the Libra = 1.012
pounds; Almude, of Lisbon = 3.7 gallons; Almude, of Oporto = 5.6 gallons ;
Alquiere = 3.6 bushels; Moio = 2.78 quarters.
ITALY.
The French metric system is used in Italy. The metre is named the
Metra; the are, Ara; the stfere, Stero; the litre, Litro; the gramme, Gramma;
the tonneau m^trique, Tonnelata de Mare. The various old weights and
measures of the different Italian States are still occasionally used.
TURKEY.
Length. — i Pike or Dri= 27 inches, divided into 24 Kerats; i Forsang
= 3.116 miles, divided into 3 Berri; the Surveyor's Pik, or the Halebi
= 27.9 inches; and 5J^ Halebis= i reed.
Surface, — The squares of the Kerat, the Pike, and the Reed. The
Feddan is an area equal to as much as a yoke of oxen can plough in a
day.
Capacity^ Dry, — ^The Rottoli.411 quarts, contains 900 Dirhems;
22 Rottols= I Killow= 7.762 gallons, or .97 bushel, the chief measure for
grain.
Liquid. — i Oka= 1.152 pints; 8 Oke= i Almud= 1.152 gallons; i Rottol
= 2.5134 pints; 100 Rottols= i Cantar = 31.417 gallons.
Weights. — ^The Oke= 2.8342 pounds, divided into 4 Okiejehs, or 400
Dirhems of 1.81 drams; i Rottolo= 1.247 pounds; 100 Rottolos= i Cantar
= 124.704 pounds.
GREECE AND IONIAN ISLANDS.
The French metric system is employed in Greece. The metre is named
the Pecheus; kilometre, Stadion; are, Stremma; litre, Litra; gramme.
Drachm^. 1% kilogrammes = i Mni; lyi Quintals = i Tolanton
i}i Tonneaux= I Tonos= 29.526 cwts.
In the Ionian Islands, whilst they were under the protection of Great
Britain (1830 to 1864), the British weights and measures were those in use,
with Italian names. The foot was named the Piede; the yard, the Jarda;
the pole, the Camaco; the furlong, the Stadio; the mile, the Miglio. The
gallon was the Gallone; the bushel, the Chilo; the pint, the Dicotile; the
pound avoirdupois, the Libra Grossa; the pound troy, the Libra Sottile.
The Talanto consisted of 100 pounds, and the Miglio of 1000 pounds.
MALTA.
In round numbers, 3^ Palmi= i yard; i Canna = 2 "/^ yards.
The Salma = 4.964 acres. Approximately, 543 Square Palmi = 400
square feet; 16 Salmi = 71 acres.
EGYPT. — LENGTH, SURFACE, ETC. 1 79
I Cubic Tratto = 8 cubic feet; 144 Cubic Palmi = 96 cubic feet; i Cubic
Canna = 543 cubic feet
Approximate weights: — 15 Oncie=i4 ounces; i Rotolo=i^ pounds;
4 Rotoli = 7 pounds; 64 Rotoli = i cwt. ; i Cantaro =175 pounds; i Quintal
= 199 pounds; 64 Cantari = 5 tons.
EGYPT. — ^Weights and Measures. — ^Tables No. 58.
I. Egyptian Measures of Length.
Pik, or cubit of the Nilometre 20.65 inches.
Pik, indigenous 22.37 „
Pik, of merchandise 25.51 „
Pik, of construction 29.53 „
6 Palms I Pik.
24 Kirats i Pik or Dr^,
4.73 Piks of construction... i Kassaba in surveying, =11.65 feet.
11. Egyptian Measures of Surface.
I Square Pik = 6.055 square feet
22.41 Square Piks i Square Kassaba = 15.07 square yards
333.33 Square Kassaba, i Feddan = .9342 acre.
III. Egyptian Measures of Capacity. *
I Kadah = 1.684 pints.
2 Kadahs i Milwah = 6.735 »
2 MDwahs I Roobah = 1.684 gallons.
2 Roobahs '. i Kelah = 3.367 „
2 Kelehs i Webek = 6.734 „
6 Webeks i Ardeb  / ^0.404 gallons, or
^ ^^«^^*^ ^ ^^^^  I 6.48 cubic feet
The Guirbah of water (a government measure) is V15 cubic metre = 66*4
litres, or 11.772 cubic feet
IV. Egyptian Measures of Weight.
I Kamhah  .746 grain.
4 Kamhahs. i Kerat
16 Kerats i Dirhem = 1.792 drachms.
24 Kerats i Mitkal.
SMitkals i Okieh.
x^: Salt's"} ^R°"°l = ^S.! pound.
100 Rottols I Kantar =98.207 pounds.
400 Dirhems i Oke = 2.728 „
36 Okes I Kantar =98.207 „
l80 WEIGHTS AND MEASURES.
MOROCCO.
Length, — ^The Tomin = 2.81025 inches; the Dra'a = 8 Tomins = 22.482
inches.
Capacity, — ^The Muhd = 3.08135 gallons; the Sai = 4 Muhds= 12.3254
gallons.
Weights, — The Uckia = 392 grains; the Rotal or Artal = 2o Uckieh =
1,12 pounds; the Kintar= 100 Rotales= 112 pounds.
Oil is sold by the Kula = 3.3356 gallons. Other liquids are sold by
weight
TUNIS.
Length, — ^The Dhrai, or Pike, is the unit of length. The Arabian Dhraa,
for cotton goods =19.224 inches; the Turkish Dhrai, for lace = 25.0776
inches; the Dhrai Endaseh, for woollen goods = 26.4888 inches.
The Mil Sah'ari = .9i49 mile.
Capacity, — For dry goods the Sai= 1.2743 pint; 12 Saa=i Hueba =
6.8228 gallons.
For liquids, the Pichoune = .4654 pint; 4 Pichounes=i Pot =1.8616
pints; 15 Pots = I Escandeau, and 4 Escandeaux=i Mill^role= 13.9623
gallons.
ARABIA.
The weights" and measures of Egypt are used in Arabia.
CAPE OF GOOD HOPE.
The standard weights and measures are British, with the excepfton of the
land measure. To some extent, the old British and the Dutch measures
are in use. The general measure of surface is the old Amsterdam Morgen^
reckoned equal to 2 acres; though the exact value is equal to 2.1 1654
acres. 1000 Cape feet are equal to 1033 British feet
INDIAN EMPIRE. — ^WEIGHTS AND MEASURES.
An Act " to provide for the ultimate adoption of an uniform system of
weights and measures of capacity throughout British India " was passed in
October, 187 1. The ser is adopted under the Act as the primary standard
or unit of weight, and is a weight of metal in the possession of the Govern
ment, equal, when weighed in a vacuum, to one kilogramme. The unit of
capacity is the volume of one ser of water at its maximum density, equiva
lent to the litre. Other weights and measures are to be multiples or sub
multiples of the ser, and of the volume of one ser of water.
The following are the weights and measures in common use in India: —
BENGAL — LENGTH, SURFACE, ETC l8l
BENGAL. — ^Weights and Measures. — Tables No. 59.
I. Bengal Measures of Length.
I Jow, or Jaub = ]^ inch.
3 Jow I Ungulee = ^ „
4 Ungulees i Moot = 3 inches.
3 Moots I Big'hath, or Span = 9 „
2 Big'haths i Hit'h, or Cubit... = 18 „
2 Hat'h I Guz = I yard.
2 Guz I Danda, orFathom = 2 yards.
^o ^^^ ^ Coss = { 15t'i.
4 Coss I Yojan = 4.5454 miles.
II. Bengal Measures of Surface.
I Square Hat'h = 2.25 square feet.
4 Square Hit'hs i Cowrie = i square yard.
4 Cowries i Gunda = 4 square yards.
20 Gundas i Cottah = 80 „
20 Cottahs I Beegah = i '^°° ^""f^ V^^' °'
\ 3306 acre.
For land measure, the following table is used for Government surveys : —
I Guz = 33 lineal inches.
3 Guz I Baus,orRod= 8^ lineal feet.
9 Square Guz i Square Rod = 68 Vx6 square feet.
400 Square Rods..... i Beegah = { 3°*S Weyards, or
III. Bengal Measures of Capacity.
The Seer is a measure common to liquids and dry goods. It is taken
at 68 cubic inches, or 1.962 pints, in volume. But it varies in different
localities. 5 Seer= i Palli, and 8 Palli= i Maund, or 9.81 gallons. The
Sooli = 3.065 bushels, and 16 Soolis = i Khahoon, or 49.05 bushels.
IV. Bengal Measures of Weight.
The Tola, or weight of a Rupee, 180 grains, is the unit of weight.
I Tola =180 grains.
5 Tolas I ChittHk =900 „
16 Chittiks I Seer = 2.057 pounds.
5 Seers i Passeeree = 10.286 „
8 Passeerees i Maund = 82.286 „
MADRAS. — Weights and Measures. — Tables No. 60.
I. Madras Measures of Length.
The English foot and yard are used. The Guz is 33 inches. The Baum
or fathom is about 6j4 feet. A NilliValli is a little under ij^ miles.
7 NailiValli = I Kadam, or about 10 miles. The following are native
measures : —
1 82 WEIGHTS AND MEASURES.
8 Torah i Vurruh = .4166 inch.
24 Vurmh i Mulakoli = 10 inphes.
4 Mulakoli i Dumna = 40 „
II. Madras Measures of Surface.
The English acre is generally known. The native me^tsures are uncer
tain. In Madras and some other districts^ the following native measures
are used : —
I Coolie = 64 square yards.
4 Ve Coolies i Ground = 266^ square yards.
24 Grounds, or ) rown***  / 6400 square yards, or
100 Coobes J (1.3223 acres.
16 Annas (each 400 yards), i Cawnie.
III. Madras Measures of Capacity.
I OUuck = .361 pint
8 OUucks I Puddee = 1.442 quarts.
8 Puddees i Mercil = 2.885 gallons.
5 Mercils i Parah =14.426 „
80 Parahs i Garce = 18.033 quarters.
This, though the legal system, is not used. The "customary" Puddee is
still in general use; it has, when slightly heaped, a capacity of 1.504 quarts.
The Mercil has a capacity of 3.0006 gallons; but, when heaped, it is equal
to 8 heaped Puddees. The Seermeasure is the most common; its cubic
contents are from 66 }4 to 67 cubic inches.
IV. Madras Measures of Weight.
I Tola = 180 grains.
3 Tolas I PoUum = 1.234 ounces.
8 PoUums I Seer = 9874 „
5 Seers i Viss = 3.086 pounds.
8 Viss I Maund = 24.686 „
20
Maunds . Candy = { ^^J J^^ J^ '^' "
In commerce, the Viss is reckoned as 3^ pounds; the Maund, 25
pounds; and the Candy, 500 pounds.
BOMBAY. — ^\Veights and Measures. — Tables No. 61.
I. Bombay Measures of Length.
I Ungulee = 9/,6 inch.
2 Ungulee i Tussoo = i^ inches. .
8 Tussoos I Vent'h =9 „
16 Tussoos I Hat'h =18 „
24 Tussoos I Guz =27 „
The Builder's Tussoo = 2.3625 inches in Bombay; and i inch in Surat
BOMBAY, CEYLON, BURMAH. 1 83
II. Bombay Measures of Surface.
34 Ve Square Hat'h... i Kutty = 93i75 square yards.
20 Kutties I Fund = 196.35 „
Pnd ^ Beegah= {39^7^fyf «'
1 20 Beegah i Chahur = 97368 acres.
In the Revenue Field Survey, the English acre is used
III. Bombay Measures of Capacity.
I Tippree= .2800 pint.
2 Tipprees i Seer = .5600 „
4 Seers i Pylee = 2.2401 pints.
16 Pylees i Parah = 4.4802 gallons.
8 Parahs i Candy = 35.8415 „
25 Pamhs I Mooda = { '""^^^S g^Uons or
•^ ( I750I quarters.
Another liquid measure is the Seer of 60 Tolas = 1.234 pints.
In timber measurement in the Bombay dockyards, a Covit or Candi =
12.704 cubic feet.
CEYLON.
The British weights and measures are used.
BURMAH.
The English yard, foot, and inch are being adopted; also the English
Measures of Capacity. Weights. — The Piakthah or Viss is 3.6 pounds, and
contains 100 Kyats of 252 grains each.
CHINA. — ^Weights and Measures. — Tables No. 62.
I. Chinese Measures of Length.
I Fun = .141 inch.
10 Fun I Tsun = i. 41 inches.
10 Tsun I Chih =14.1 „
10 Chih I Chdng= 11.75 ^"^^t.
10 Ching I Yin =39.17 yards.
The Chih of 1 4.1 inches is the legal measure at all the ports of trade.
At Canton, the values of the Chih are as follows : —
Tailor's Chih 14.685 inches.
Mercer's Chih (wholesale) 1 4. 66 to 1 4. 7 2 4 inches.
Mercer's Chih (retail) ^4*37 to 14.56 „
Architect's Chih 12.7 inches.
At Pekin there are thirteen different Chihs.
1 84 WEIGHTS AND MEASURES.
Distance,
5 Fun I Li = .486 inch.
10 Li )^Chih = .405 foot.
5 Chih I Pii = 4.05 feet.
360 Pii I Li =486 yards.
250 Lf I Tii (or Degree) = 69 miles.
II. Chinese Measures of Surface.
25 Square Chih i Pii or Kung= 3.32 square yards
60 Kung I Kish =199.47
4 Kish I Mau =79789
100 Mau I King = 16,485 acres.
The chief land measure is the Mau, than which smaller areas are
expressed decimally.
III. Chinese Measures of Capacity. {Dry Measure,)
I Koh = .0113 gallon.
5 Koh >^Shing = .0565 „
10 Koh I Shing = .113 „
10 Shing I Tau =1.13 gallons.
Liquids are measured by vessels containing definite weights, as i, 2, 4^
and 8 Taels; also large earthen vessels holding 15, 30, and 60 Catties.
See Table of Weights.
IV. Chinese Measures of Weight.
I L^ang or Tael = i '/a ounces.
16 L^ang I Kin or Catty = i V3 pounds.
100 fcin I Tan or Pecul = [ ^3333 Pounds, or
( 1. 19 cwts.
COCHINCHINA.
Length, — The Thuoc, or cubit, 19.2 inches, is the chief unit of measure
of length. It varies considerably for different places. The Li or mile is
486 yards; 2 Li make i Dam; and 5 Dam make i league =2.761 miles.
Surface, — 9 Square Ngu make i Square Sao = 64 square yards. 100
Square Sao make i Square Mao = 6400 square yards, or 1.32 acres.
Weights, — The smallest weight is the Ai = .0000006 grain. The weights
ascend by a decimal scale, until 10,000,000,000 Ai are accumulated =
I Nen = .8594 pound. The greatest weight is the Quan = 6875^ pounds.
Capacity for Grain, — i Hao = 6»/g gallons. 2 Hao = i Shita=i2 4/j
gallons.
PERSIA.
Length, — The Gereh = 2^ inches; 16 Gerehs= i Zer=38 inches. The
Kadam or Step = about 2 feet; 12,000 Kadam = i Fersakh = about 4)^ miles.
PERSIA, JAPAN. 185
Surfcue and Cubic Measures, — ^These are the squares and cubes of the
lengths.
Capacity (Dry Goods). — The Sextario = .o7236 gallon. 4 Sextarios =
I Chenica; 2 Chenicas= i Capicha; 3^ Capichas= i CoUothun; 8 Collo
thun = I Artata= 1.809 bushels.
liquids are sold by weight
Weights, — ^The Miscal = 7i grains; 16 Miscals=i Sihr; 100 Miscals =
I Ratal = 1.014 pounds; 40 Sihrs = i Batman (Maund) = 6.49 pounds; 100
Batman (of Tabreez)= i Karwar = 649.i42 pounds.
JAPAN. — ^Weights and Measures. — Tables No. 63.
I. Japanese Measures of Length.
I Rin = .012 inch.
10 Rin I Boo = .120 inch.
10 Boo I Sun = 1.20 inches.
10 Sun I Shiaku = i foot.
3 Shiaku ^ Ken = i yard.
6 Shiaku i Ken  2 yards.
60 Ken I Chu = 120 yards.
_ J 4320 yards, or
2.454 miles.
36 Chu., I Ri = I
Rough timber is sold by the YamaKenZau = d^ Sun. Cloth is measured
by the Shiaku of 15 inches, with decimal submultiples.
II. Japanese Measures of Surface.
I Po = 4 square yards.
30 Po I Is'she = 120 square yards.
I o Is'she I It'tau = 1 200 square yards.
10 Iftau. I Ifchoe = ( "°°° T"^ y^''*'' °'"
I 2.4793 acres.
The square Ken is the unit of square measure, equal to 4 square yards,
III. Japanese Measures of Capacity.
* Dzoku = .0000328 pint.
10 Dzoku I Ke = .000328 pint.
10 Ke I Sat = .000328 pint.
10 Sats I Sai = .00328 pint.
10 Sal I Shiaku = .03283 pint.
10 Shiaku i Goo = .3283 pint
10 Goo I Shoo = .4104 gallon.
10 Shoo I To = 4.104 gallons.
xo To I Koku =41.04 gallons.
1 86 WEIGHTS AND MEASURES.
IV. Japanese Measures of Weight.
I Mo = .027 grain.
10 Mo I Rin = .2701 grain.
10 Rin I Fun = 2.701 grains.
10 Fun I Noihme = 27.006 grains.
4 Nomme i Riu = 108.026 grains.
40 Riu I Kiu = .6173 pound.
JAVA.
Length. — The Duim=i.3 inches. 12 Duims=i foot. The Ell =
27.08 inches.
Surface, — The Djong of 4 Bahu = 7.015 acres.
Capacity, for rice and grain. — The measures are in fact measures of
definite weights, i sack = 61.034 pounds; 2 sacks =1 Pecul; 5 Peculs
= I Timbang = 5.45 cwts.; 6 Timbang= i Coyaii = 32.7 cwts. For Hquids:
The Kan = .328 gallon; 388 Kans=: 1 Leager= 127.34 gallons.
Weights. — The Tael = 593.6 grains; i6Taels=^i Catty = 1.356 pounds:
100 Catties = i Pecul 135.63 pounds.
UNITED STATES OF AMERICA.
Length, — The measures are the same as those of Great Britain.
In Land Surveying, the unit of measurement is the chain, and it is deci
mally subdivided.
In City Measurements, the unit is the foot, and it is decimally subdivided.
In Mechanical Measurements, the unit is the inch, and it is divided into
a hundred parts.
Surface, — The measures are the same as those of Great Britain.
Capacity, — The measures of capacity for dry goods and for liquids are the
same as the old English measures. 'Fhe standard U. S. gallon is equal to
the old English wine gallon, or 231 cubic inches; it contains 8^ pounds
of pure water at 62° F.
Dry Measure, — Table No. 64.
I gill. = .96945 imperial gill.
4 gills I pint = .96945 imperial pint
2 pints I quart ~ 1.9388 „ pints.
4 quarts i gallon = .96945 „ gallon.
2 gallons I peck = 1.9388 „ gallons.
4 pecks I bushel = 96945 „ bushel.
4 bushels I coomb =38777 „ bushels.
2 coombs I quarter = .96945 „ quarter.
5 quarters i wey or load =4.8472 „ quarters.
2 weys I last =96945 „ quarters.
For the Wine and Spirit Measures, and the Ale and Beer Measures, see
the Old Measures of Great Britain, page 139.
I cord of wood =128 cubic feet = (4 feet x 4 feet x 8 feet).
Weights, — The Weights are the same as those of Great Britain. (See
page 140.)
BRITISH NORTH AMERICA, ETC. 1 8/
There are, in addition, the Quintal or Centner of loo pounds; and the
New York ton of 2000 pounds, which is also used in most of the States,
The old hundredweight and ton are for the most part superseded by the
quintal and the New York ton.
The French metric system of weights and measures has been legalized
concuirendy with the old system.
BRITISH NORTH AMERICA.— WEIGHTS AND MEASURES.
Until the 23d May, 1873, the standard measures of length and surface,
and the weights, were the same as those of Great Britain; whilst the
measures of capacity were the old British measures for dry goods, for wine,
and for ale and beer. At the abovenamed date a new and uniform system
of weights and measures came into force, in which the imperial yard, pound
avoirdupois, gallon, and bushel, became the standard units, and the
imperial system was adopted in its integrity, with two important exceptions :
that the hundredweight of 112 pounds, and the ton of 2240 pounds were
abolished; and the hundredweight was declared to be 100 pounds, and the
ton 2000 pounds avoirdupois, — thus assimilating the weights of Canada to
those of the United States.
The French metric system of weights and measures has been made
permissive concurrently with the standard weights and measures.
MEXICO.
The weights and measures are the old weights and measures of Spain.
CENTRAL AMERICA AND WEST INDIES.
WEST INDIES (British).
The weights and measures are the same as those of Great Britain.
CUBA.
The old weights and measures of Spain are in general use. For engineer
ing and carpentry work the Spanish, English, and French measures are in
use. The French metric system of weights and measures is legalized, and
is used in the customs departments.
GUATEMALA AND HONDURAS.
The weights and measures are the old weights and measures of Spain.
BRITISH HONDURAS.
In British Honduras, the British weights and measures are in use.
COSTA RICA.
The old weights and measures of Spain are in general use. But the
introduction of the French metric system is contemplated.
l88 WEIGHTS AND MEASURES.
ST. DOMINGO.
The old Spanish weights and measures are in general use. The French
metric system is coming into use.
SOUTH AMERICA.
COLOMBIA.
The French metric system was introduced into the Republic in 1857,
and is the only system of weights and measures recognized by the govern
ment. In ordinary commerce, the Oncha, of 25 lbs., the Quintal, of
100 lbs., and the Carga, of 250 lbs., are generally used. The libra is
1. 102 pounds. The yard is the usual measure of length.
VENEZUELA.
The system and practice are the same as those of Colombia.
ECUADOR.
The French metric system became the legal standard of weights and
measures on the ist January, 1858.
GUIANA.
In British Guiana, the weights and measures are those of Great Britain.
In French Guiana or Cayenne, the ancient French system is practised In
Dutch Guiana, the weights and measures of Holland are employed.
BRAZIL.
The French metric system, which became compulsory in 1872, was
adopted in 1862, and has since been used in all official departments. But
the ancient weight§ and measures are still partly employed. They are, with
some variations, those of the old system of Portugal.
Length, — ^The Line = .09 11 inch, and is divided into tenths. The PoUe
gada = 1.0936 inches. The Pd = 1 3. 1 236 inches, or ^j^ metre. The Vara =
1.2 1 5 yards; and ij^ Varas = the geometrical pace =1.8227 yards. The
Milha= 1.2965 miles; and 3 Milhas = i Legoa = 3.8896 miles.
6 yards are reckoned equal to 5 Varas.
Surface,
64 Square Pollegadas... i Square Palmo = .5315 square foot
25 Square Palmos i Square Vara =1.4766 square yards.
4 Square Varas i Square Braga = 5. 9063 „
4840 Square Varas i Geira = 1.4766 acres.
Capacity {Dry Goods), — The Salamine = .38o8 gallon; 2 Salamines =
\ Oitavo; 2 Oitavo = i Quarto; 4 Quartas=i Alqueiro = .38o8 bushel;
4 Alqueiras = i Fangas; 15 Fangas = i Moio = 2.8560 quarters.
Liquids, — The Quartilho = .6i4i pint; 4 Quartilhos= i Canada; 6 Cana
das = I Pota or Cantaro; 2 Potas = i Almuda = 3.6846 gallons.
PERU, CHILI, BOLIVIA, ETC. 1 89
Weights. — The Arratel= 1.0119 pounds, is divided into 16 Ongas, and
then into 8 Oitavos. 32 Arratels=i Arroba; 4 Arrobas = i Quintal =
129.5181 pounds; and 13 J^ Quintals = 1 Tonelada= 15.6116 cwts.
There is also the Quintal of 100 Arratels; Ships' freight is reckoned by
the English ton = 70 Arrobas.
PERU.
The French metric system was established in i860, but is not yet gener
ally used. The weights and measures in common use are : — ^The ounce =
1.014 ounce; the Libra=i.oi4 pound; the Quintal = 101.44 pounds; ^^
Arroba =25.36 pounds, or 6.70 gallons; the gallon = .74 imperial gallon;
the Vara = .92 7 yard; the square Vara = .85 9 square yard.
CHILL
The French metric system has been legally established; but the old
weights and measures are still in general use. These are the same as those
of Peru.
BOLIVIA.
The weights and measures are the same as the old weights and measures
of Peru and Chili.
ARGENTINE CONFEDERATION.
The French metric system has recently been established. The old
weights and measures are commonly used: — th^ Castilian standards of the
old Spanish system. The Quintal = loi. 4 pounds; the Arroba = 25.35
pounds; the Fanega=i.5 bushels.
URUGUAY.
The French metric sjrstem was established in 1864. The old weights
and measures are the same as those of the Argentine Confederation. The
weights and measures of Brazil are in general use.
PARAGUAY.
The weights and measures are the same as the old ones of the Argentine
Confederation.
AUSTRALASIA.
In New South Wales, Queensland, Victoria, South Australia, West
Australia, Tasmania, and New Zealand, the legal weights and measures are
the same as those of Great Britain. But the old British measures of
capacity are also much used.
In land measurement, a "section" is an area equal to 80 acres.
MONEY.
GREAT BRITAIN AND IRELAND.
Coins.
Material.
Weight.
Grains.
j{d, farthing bronze.
}id. halfpenny do.
4 farthings i penny do.
3//. threepenny piece silver.
4i/. groat, or fourpenny piece do.
6d. sixpence do.
12 pence i shilling do.
2 shillings i florin do.
2j^x. I halfcrown do.
lox. I halfsovereign gold..
20s, I sovereign, or pound sterling do.
• 43 750
. 87.500
.145833
. 21.818
. 29.091
• 43.^36
. 87.273
.174.545
.218.182
. 61.6372
.123.2745
The bronze coins are made of an alloy of copper, tin, and zinc; the
silver coins contain 92 J^ per cent, of fine silver, and Tj4 per cent of alloy;
the gold coins, 91^ per cent, of fine gold, and 8^ per cent, of alloy.
The Mint price of standard gold is ;£'3, ijs, ioJ4//. per ounce.
One pound weight of silver is coined into 66 shillings. The intrinsic
value of 22 shillings is equal to £1 sterling.
The intrinsic vdue of 480 pence is equd to £1 sterling.
FRANCE.— Money.
Vxoo firanc .
V50 franc
Vao franc
Copper,
Coins. Weight. Value in English Money.
Grammes. £ S, a,
1 centime i o
2 centimes 2 o
5 centimes (j^«) 5 o
'/xo franc 10 centimes (gr^jj^w)... 10 .0
o
o
o
o
xo
'A
I
2
5
franc,
franc,
franc.
Silver,
20 centimes i...
50 centimes 2.5.
100 centimes 5
.0
.0
.0
o 2
o aM,
o g}4
francs 10
francs 25
more exactly 9.524//.
.0
.0
I
3
7
GERMANY, HANSE TOWNS. I9I
GoU.
Grammes. JZi ^' "•
5 francs. 1*61290 o 3 11^
10 francs. 3*22580 o 7 ii^
20 francs (Nj^)oleon)... 6*45161 (99*56 grains)...o 15 10^
50 francs 16*12902 i 19 8^5
100 francs 32*25805 3 19 4 4/,^
The English value is calculated at the rate of 25 francs 20 centimes to
j£i. The bronze coins consist of an alloy of 95 parts of copper, 4 of tin,
I of zinc. The standard fineness of the gold pieces, and of the silver
5franc pieces is 90 per cent., with 10 per cent of copper; of the other
alver coins, 83.5 per cent.; and of the bronze coins, 95 per cent.
GERMANY.— Money.
The following system of currency was established throughout the German
Empire in 1872: —
English Value.
s, d.
I Pfennig = o .1175
ID Pfennig i Groschen = o 1.175
10 Groschen i Mark = o iij^
10 Marks (gold).. = 9 9^
20 Marks (gold)... = 19 7
The 20mark gold piece weighs 122.92 grains, and the standard fineness
of the gold pieces is 90 per cent of gold.
Before 1872, accounts were reckoned in the following currency in North
Germany : —
s, d,
12 Pfennig i Silbergroschen = i i '/s
30 Silbergroschen i Thaler = 30
In South Germany: —
4 Pfennig i Kreutzer = o ^
60 Kreutzers. i Florin = 18
HANSE TOWNS.— Money.
The monetary system is that of the German Empire.
Hamburg. — ^According to the old monetary system, in which silver was
the standard, 12 Pfennig =1 Schilling = ^ltd.\ and 16 Schillings =1 Mark
= i3>i^.
Bremen. — Old system: — 5 Schmaren= i Groot = "/aorfl; and 71 Groots =
I Rixdollar=3J. z^li^ The Rixdollar, or Thaler, was a money of
account
Luhec. — ^The old system was the same as that of Hamburg, and, in
addition, 3 Marks = i Thaler = 3 j. 4^/.
192 MONEY.
AUSTRIA.— Money.
J. d,
I Kreutzer (copper) o '/j
4 Kreutzers (do.) o 4/^
10 Kreutzers (silver) o 2^
20 Kreutzers i^o.) o 4^^
^ Florin (do.) o sH'
1 Florin (do.) i 11^
2 Florins (do.) 3 11}^
4 Florin piece (gold) 7 11
8 Florin piece (do.) 15 10
100 Kreutzers make i Florin.
The 4florin gold piece weighs 49.92 grains, and the standard of fineness
is 90 per cent of gold.
RUSSIA. — Money.
I Copeck
1 00 Copecks I Silver Rouble .
s.
d.
=
.38
—
3
2
The copper coins are pieces of ^, ^, i, 2, 3, 5 Copecks. The silver
coins are pieces of 5, 10, 15, 20, 25 Copecks, the Half Rouble, and the
Rouble; the gold coins are the Threerouble piece, the Half Imperial of
five Roubles, and the Imperial of 10 Roubles. The 5rouble gold piece
weighs 1 01 grains, and the standard of fineness is 91^ per cent of gold.
Paper currency: — i, 3, 5, 10, 25, 50, 100 Roubles.
HOLLAND.— Money.
s, d,
I Cent = o '/j
100 Cents I Guilder or Florin = i 8
BELGIUM.— Money.
The monetary system is exactly the same as that of France.
DENMARK.— Money.
s, d,
I Skilling = o .2745
16 Skillings i Mark = o 4.392
96 Skillings, or 6 Marks i Rigsdaler, or Daler = 22 7/ao
SWEDEN. — Money.
s, d,
I Ore = o .133
100 Ore I Riksdaler = i i^
NORWAY.— Money.
s, d.
I Skilling = o .444
24 Skillingen i Ort or Mark = o lo^
5 Ort I SpeciesDaler = 4 $)i
SWITZERLAND, SPAIN, ETC I93
SWITZERLAND.— Money.
The monetary system of Switzerland is the same as that of France. The
Centime is called a Rappe.
SPAIN. — Money.
d.
I Centimo = 95
100 Centimos i Peseta = i franc, or 9)^
The bronze coins are pieces of i, 2, 5, and 10 centimos. The silver
coins are pieces of 20 and 25 centimos, and i, 2, and 5 pesetas. The gold
coins are pieces of 5, 10, 20, 25, 50, and 100 pesetas. The piece of
5 pesetas is y, iij4^., English value. The 25 peseta piece is 19J. 9^^.,
English value.
The old monetary system was based on the RealVellon, 2j^^. English
value; it was the 20th part of the Silver Hard Dollar^ 4^. 2//. English value,
and of the Gold Dollar or Coronilla. The Duro was identical with the
American Dollar.
PORTUGAL.— Money.
The unit of account is the Rei, of which 185^ Reis make i penny; and
4500 Reis make i sovereign. The Milreis is 1000 Reis, ^r. syid, English
value. The Corda is the heaviest gold coin, of 10,000 Reis, JQ2, ^r. 5^//.
English value, and weighs 17.735 grammes.
ITALY. — Money.
d,
I Centime = .95
100 Centimes i Lira = i franc, or 9^
Copper coins are pieces of i, 3, and 5 Centimes; silver coins, 20 and
50 Centimes, and i, 2, and 5 Lire; gold coins, 5, lo, 20, 50, and 100 Lire.
lliese coins are the same in weight and fineness as the coins of France.
TURKEY. — Money.
s, d,
I Para = o 7,8.5
40 Paras i Piastre = o 2.16
100 Piastres i Medjidie, or Lira Turca = 18 o
The Piastre is roughly taken equal to 2d. sterling.
GREECE AND ;ONIAN ISLANDS.— MONEY.
icx) Lepta i Drachma = i franc, or 9 J^//.
The currency of Greece is the same as that of France.
In the Ionian Islands, whilst they were under British protection (1830
1864), accounts were kept by some persons in Dollars, of 100 Oboli = 4f. 2//.;
by others in Pounds, of 20 shillings, of 12 pence, Ionian currency; the
Ionian Pound being equal to 20s. g,6d sterling. By other persons accounts
were kept in Piastres of 40 Paras =^ 2 */^^,
13
194 MONEY.
MALTA. — Money.
I Grano
20 Grani i Taro
12 Tari I Scudo
Or,
60 Piccioli I Carlino
9 Carlini i Taro
12 Tari i Scudo
s.
d.
v»
I?^
I
8
.185
x?^
I
8
British money is in general circulation. The Sovereign = 12 Scudi; the
Shilling = 7 Tari 4 Gram.
EGYPT. — Money.
£ J. //.
I Para = 00 .0615
40 Paras i Piastre (Tariff) = o o 2.461
100 Piastres i Egyptian Guinea = i o 6.84
5 Egyptian Guineas... I Kees, or Purse = 52 10.2
1000 Purses I Khuzneh, or Treasury = 5142 10 o
97.22 Piastres i English Sovereign.
The Egyptian guinea weighs 132 grains, and the standard of fineness is
87^ per cent of gold.
Two piastres (current) are equal to one piastre (tariff).
MOROCCO.— Money.
I Flue
24 Flues I Blankeel =
4 Blankeels i Ounce =
10 Ounces i Mitkul =
TUNIS. — Money.
I Fel =0 3S/,88
3 Fels I Karub = o 3S/^
16 Karubs i Piastre = o S ^/e
ARABIA. — Money.
s. d
80 Caveers i Piastre or Mocha Dollar =35
CAPE OF GOOD HOPE. — MoNEY.
Public accounts are kept in English money; but private accounts are
often kept in the old denominations, as foUows : —
I Stiver =
6 Stivers i Schilling =
8 Schilling i RixdoUar =
The Guilder is equal to 6d.
s.
//.
37/960
3V40
37
3
I
s.
d
3/8
^H
I
6
INDIAN EMPIRE, CHINA, ETC 1 95
I14DIAN EMPIRE.— Money.
Throughout India, accounts are kept in the following moneys: —
s, d.
I Pie = o o^ nominal value.
12 Pies I Anna. = o ij4 do.
16 Annas i Rupee = 20 do.
The intrinsic value of the Rupee is is. io}id.; it weighs 180 grains.
The English Sovereign is equal to 10 Rupees 4 Annas.
I Lac of Rupees = 100,000 rupees = ^10,000.
I Crore of Rupees =100 lacs = ;;^i, 000,000.
In Ceylon, the Rupee is divided into 100 Cents.
The gold coin, Mohur, is equal to 15 rupees; it weighs 180 grains, and
the standard fineness is 91.65 per cent of gold.
CHINA.— Money.
J. a.
I Cash (Le) = o 7/,oo
10 Cash I Candajreen (Fun) = o y/,©
10 Candareens i Mace (Tsien) =0 7
10 Mace iTael(Leang) = 5 10
COCHINCHINA.— Money. .
s. a,
I Sapek, or Dong, or Cash = o «/i8
60 Sapeks i Mas, or Mottien = o 3^
10 Mas. I Quan, or String. = 2 9^
PERSIA.— Money. .
s, a.
I Dinar = o '/a©
50 Dinars i Shahi =0 ^
20 Shahis i Keran = o iij^
10 Kerans i Toman = 9 3J6
JAPAN.— Money.
10 Rin. I Sen = J^
100 Sen I Yen =42
There are gold coins of the value of i, 2 and 5 yen, with a standard
fineness of 90 per cent The 5yen piece weighs 128.6 grains. The silver
yen weighs 416 grains, with the same standard of fineness.
JAVA. — Money.
The money account of Java is the same as that of Holland.
UNITED STATES OF AMERICA.— MoNEY.
s. d,
I Cent =0 }i
10 Cents I Dime.... =05
100 Cents I Dollar. =42
196 MONEY.
CANADA.— BRITISH NORTH AMERICA.— MoNEY.
s. d.
I Mil = o Vao sterling.
10 Mils I Cent = o J^ do.
100 Cents I Dollar = 4 i}( do.
4 Dollars = 20 o currency.
Or,
I Penny currency = o ^ sterling.
12 Pence i Shilling do = o 94/^ do.
20 Shillings i Pound do = 16 5^ do.
The Dollar of Nova Scotia, New Brunswick, and Newfoundland, is equal
to 4J. 2d. sterling. In the Bermudas, accounts are kept in sterling money.
MEXICO.— Money.
Accounts are kept in dollars of 100 cents. The dollar is equal to 4s, 2d.
sterling.
CENTRAL AMERICA AND WEST INDIES.— MoNEY.
WEST INDIES (British).
Accounts are kept in English money; and sometimes in dollars and
cents. I dollar = 4r. 2d,
CUBA. — Money.
The moneys of various nations were in circulation before the current
war (1875). But the principal silver currency was the 10 cent and 5 cent
pieces of the United States. The gold currency consists of the Ounce, of
the value of 16 dollars, }4 ounce, jounce, }i ounce.
GUATEMALA, HONDURAS, COSTA RICA.
The mone)rs of account are the same as those of Mexico.
ST. DOMINGO.
Accounts are kept in current dollars (called Gourde) and cents. The
cent= Vsa^'i ^^d ^00 cents = i dollar = 3^//.
SOUTH AMERICA. — MoNEY.
COLOMBIA, VENEZUELA, ECUADOR.
The moneys of account are, the Centavo= J^^.; and 100 Centavos =
I Peso = 4J'. 2d,
GUIANA.
In British Guiana the dollar of 4s, 2d, is used, divided into 100 cents.
In French Guiana, French money is used. In Dutch Guiana, the money of
Holland is used.
BRAZIL, PERU, ETC. 1 97
BRAZIL. — Money.
s. d.
I Rei  o «Vioo
looo Reis '. I Milreis  2 3
PERU. — Money.
s. d,
I Centesimo =0 .37
1 00 Centesimos i Dollar, or Peso =31
CHILI. — Money.
J. d/
I Centavo =0 .45
100 Centavos i Dollar, or Peso =39
BOLIVIA.
I Centena =0 .37
100 Centenas i Dollar =31
ARGENTINE CONFEDERATION.
I Centesimo =0 .25
100 Centesimos i Dollar, or Patercon =21
URUGUAY.
I Centime =0 oj4
100 Centimes i Dollar =42
PARAGUAY.
I Centena =0 .37
100 Centenas i Dollar =31
AUSTRALASIA.
Accounts are kept in pounds, shillings, and pence sterling.
WEIGHT AND SPECIFIC GRAVITY.
The specific gravity, or specific weight of a body, is the ratio which the
weight of the body bears to the weight of another body of equal volume
adopted as a standard for comparison of the weights of bodies. For solids
and liquids, pure water at the mean temperature 62** F., is adopted as the
standard body for comparative weight. For gases, dry air at 32° F., and
under one atmosphere of pressure, or 14.7 lbs. per square inch, is the body
with which they are compared.
The specific gravity of bodies is found by weighing them in and out of
water, according to the following rules.
Rule i. — To find the specific gravity of a solid body heavier than water.
Weigh it in pure water at 62® F., and divide its weight out of water by the
loss of weight in the water. The quotient is the specific gravity.
Note, — The loss of weight in water is the difference of the weight in air
and the weight in water, and it is equal to the weight of the quantity of
water displaced, which is equal in volume to the body.
Rule 2. — To find the specific gravity of a solid body lighter than water.
Load it so as to sink it in pure water at 62'' F., and weigh it and the load
together, out of water, and in water; weigh the load separately in and out
of water; deduct the loss of weight of the load singly from that of the
combined body and load ; the remainder is the loss of weight of the body
singly, by which its weight out of water is to be divided. The quotient is
the specific gravity.
Rule 3. — To find the specific gravity of a solid body which is soluble in
water. Weigh it in a liquid in which it is not soluble; divide the weight
out of the liquid by the loss of weight in the liquid, and multiply by the
specific gravity of the liquid. The product is the specific gravity of the
body.
Rule 4. — To find the specific gravity of a liquid. Weigh a sblid body in
the liquid and in water, as well as in the air, and divide the loss of weight
in the liquid by the loss of weight in water. The quotient is the specific
gravity.
Rule 5. — To find the weight of a body when the specific gravity is given.
Multiply the specific gravity by
MULTIPLIES. WEIGHT OP
62.355 (t^^ weight in pounds of a cubic foot of
piure water at 62° F.) = i cubic foot, in lbs.
1683.60 =1 cubic yard, in lbs.
15.0 =1 „ incwts.
.75 ^i „ intons.
WEIGHT AND SPECIFIC GRAVITY. 1 99
Note, — ^As one cubic foot of water at 62° F. weighs about 1000 ounces
(exactly 997.68 ounces), the weight in ounces of a cubic foot of any other
substance will represent, approximately, its specific gravity, supposing
water =1000.
If the last three places of figures be pointed off as decimals, the result
will be the specific gravity approximately, water being = i.
In France, the standard temperature for comparison of the density of
bodies, and the determination of their specific gravities, is that of the
maximum density of water, — about 4° C, or 39°.! F., for solid bodies; and
32' F., or 0° C, for gases and vapours, under one atmosphere or .76 centi
metres of mercury. In practice, it is usual to adopt the cubic decimetre or
litre as the unit of volume, since the cubic decimetre of distilled water, at
4* C. weighs, by the definition, i kilogramme. Consequently the specific
gravity of a body is expressed by the weight in kilogrammes of a cubic
decimetre of that body.
The densities of the metals vary greatly. Potassium and one or two
others are lighter than water. Platinum is more than twenty times as
heavy. Lead is over eleven times as heavy; and the majority of the useful
metals are from seven to eight times as heavy as water.
Stones for building or other purposes vary in weight within much
narrower limits than metals. With one exception, they vary from basalt and
granite, which are three times the weight of water, to volcanic scoriae which
are lighter than water. The exception referred to is barytes, which is con
spicuously the heaviest stone, being 4j^ times as heavy as water. The
sulphate of baryta is known as heavy spar.
Amongst other solids, flintglass has three times the weight of water; clay
and sand, twice as much; coal averages one and a half times the weight of
water; and coke from one to one and a half times. Camphor has about
the same weight as water.
Of the precious stones, zircon is the heaviest, having four and a half
times the weight of water; garnet is four times as heavy, diamond three
and a half times as heavy, and opal, the lightest of all, has just twice the
weight of water.
Peat varies in weight from onefifth to a little more than the weight of
water.
The heaviest wood is that of the pomegranate, which has one and a third
times the weight of water. English oak is nearly as heavy as water, and
heart of oak is heavier; the densest teak has about the same weight as
water; mahogany averages about threefourths, elm over a half, pine from
a half to threefourths, and cork onefourth of the weight of water. Of the
colonial woods, the average of 22 woods of British Guiana weighs 74 per
cent, of the weight of water; of 36 woods of Jamaica, 83 per cent.; and of
18 woods of New South Wales, 96 per cent
Woodcharcoal in powder averages one and a half times the weight of
water; in pieces heaped, it averages only twofifths. Gunpowder has about
twice the weight of water.
Of animal substances, pearls weigh heaviest, two and threequarter times
the weight of water; ivory and bone twice, and fat over ninetenths the
weight of water.
Of vegetable substances, cotton weighs about twice as much as water;
guttapercha and caoutchouc nearly th^ §apa^ weight ^ waiter,
200 WEIGHT AND SPECIFIC GRAVITY
Mercury, the heaviest liquid at ordinary temperatures, has over thirteen
and a half times the weight of water; and bromine nearly three times the
weight The water of the Dead Sea is a fourth heavier, and ordinary sea
water two and a half per cent, heavier than water; whilst oliveoil is about
onetenth lighter, and pure alcohol and woodspirit a fifth lighter than
water.
Turning to gaseous bodies, water at 62** F. has 772.4 times the weight of
air at 32° F., under a pressure of one atmosphere; and the specific gravity
of air at 32° F. is .001293, that of water at 62*^ F. being = i. Oxygen gas
weighs a tenth more than air, gaseous steam weighs only fiveeighths of air,
and hydrogen, the most perfect type of gaseity, has only seven per cent, of
the weight of air. Water has upwards of 11,000 times the weight of
hydrogen.
One pound of air at 62° F. has the same volume as a ton of quartz.
The following Tables, Nos. 65 to 69, contain the weights and specific
gravities of solids, liquids, and gases and vapours. The specific gravities
have been derived from the works of Rankine, Ure, Wilson, Claudel, and
Peclet, Delabfeche* and Playfair, Fowke, and others whose names are men
tioned in the body of the tables. Columns containing the bulks of
bodies have been added to the tables.
The specific gravity of alloys does not usually follow the ratios of those
of their constituents; it is sometimes greater and sometimes less than the
mean of these. Ure gives the specific gravities of some alloys of copper, tin,
zinc, and lead, examined by Crookewitt. The following are the specific
gravities of the alloys, as ascertained by Crookewitt; and, for the purpose
of comparison, they are preceded by the specific gravities of the particular
samples of the elementary metals employed.
SPECIFIC GRAVITY.
Copper 8.794
Tin 7.305
Zinc 6.860
Lead ii3S4
Alloys: — Copper 2, tin 5 7«6S2
Copper I, tin i 8.072
Copper 2, tin i 8.512
Copper 3, zinc 5 7.939
Copper 3, zinc 2 8.224
Copper 2, zinc i 8.392
Copper 2, lead 3 10753
Copper I, lead i io375
Tin I, zinc 2 7.096
Tin I, zinc I 7ii5
Tin 3, zinc i 7.235
Tin I, lead 2 9*965
Tin I, lead I 9*394
Tin 2, lead I 9.025
The following binary alloys have, on the one side, a density greater than
the mean density of their constituents; and, on the other side, a density
less than the mean density of the constituents.
OF METALS AND ALLOYS.
20 1
Alloys liaving a density greater than the mean.
Gold and zinc.
Gold and tin.
Gold and bismuth.
Gold and antimony.
Gold and cobalt
Silver and zinc.
Silver and lead.
Silver and tin.
Silver and bismuth.
Silver and antimony.
Copper and zinc.
Copper and tin.
Copper and palladium.
Copper and bismuth.
Lead and antimony.
Platinum and molybdenum.
Palladium and bismuth.
Alloys having a density less than the mean.
Gold and silver.
Gold and iron.
Gold and lead.
Gold and copper.
Gold and iridium.
Gold and nickel.
Silver and copper.
Iron and bismuth.
Iron and antimony.
Iron and lead.
Tin and lead.
Tin and palladium.
Tin and antimony.
Nickel and arsenic.
Zinc and antimony.
202
VOLUME, WEIGHT, AND SPECIFIC GRAVITY
TABLE No. 65.— VOLUME, WEIGHT, AND SPECIFIC GRAVITY
OF SOLID BODIES.
FAMILIAR METALS.
Platinum
Gold
Mercury, fluid
Lead, milled sheet
Do. wire :
Silver
Bismuth
Copper, sheet
Do. hammered
Do. wire
Bronze: — 84 copper, 16 tin, gun metal
83 » 17 » «
81 » 19 i> «
79 „ 21 „ millbearmgs
35 » 65 „ small bells
21 » 79 V »
15 „ 85 „ speculum metal...
Nickel, hammered
Do. cast
Brass: — cast
75 copper, 25 zinc, sheet
66 „ 34 n yellow
60 „ 40 „ Muntz's metal, ...
Brass, wire
Manganese
Steel : — Least and greatest density
Homogeneous metal
Blistered steel
Crucible steel
Do. average '.
Cast steel,
Do. average
Bessemer steel
Do. average
M ean for ordinary calculations
Iron, wrought : — Least and greatest density...
Common bar
Puddled slab
Various — Irons tested by Mr. Kirkaldy
Do. average
Common Tails
Do. average
Yorkshire iron bar
Lowmoor plates, ii to 3 ins. thick....
Beale's rolled iron
Pure iron (exceptional), by electro )
deposit (Dr. Percy) )
Mean, for ordinary calculations
Weight of one
cubic foot
pounds.
.. 1342 ...
I2CO
849 ...
712
704 ...
655
617 ...
549
.. 556 ...
554
534 ...
528
.. 520 ..•
544
503 ...
461
465 ...
541
516 ...
505
527 ...
518
511
533
499 ...
435 to 493
493 ...
488
..488 to 490...
489
489 to 489.5
489.3
..489 to 490...
489.6
489.6 ...
466 to 487
471 ...
460.5 to 474
..468 to 486...
477
..466 to 476...
470
484 ...
487
476 ...
Specific Gfavity.
Watcr= I.
508
480
... 21.522
19.245 .
... 13.596
II.418
... 11.282
10.505
9.90
8.805
... 8.917
8.880
... 8.56
8.46
8.46
8.73
... 8.06
7.39
... 7.45
8.67
... 8.28
8.10
... 8.45
8.30
8.20
8.548
,.. 8.00
7.729 to 7.904
.. 7.904
7.823
7.825 to 7.859
7.842
7.844 to 7.851
7.848
7.844 to 7.857
7.852
.. 7.852
7.47 to 7.808
.. 7.55
7.53 to 7.60
.. 7.5 to 7.8
7.65
7.47 to 7.64
7.54
.. 7.758
7.808
.. 7.632
8.140
7.698
OF SOLID BODIES.
203
Familiar Metals {continued).
Iron, cast: — Least and greatest density
White
Gray ;
Eglinton hotblast, ist melting...
2d do. . . .
14th do. ...
Rennie
Mallett
Mean, for ordinary calculations..
Tin
Zinc, sheet
Do. cast
Antimony
Aluminium, wrought
Do. cast
Magnesium
OTHER METALS.
Iridium
Uranium
Tungsten
Thallium
Palladium.....
Rhodium
Osmium
Cadmium
Molybdenum
Ruthenium...
Cobalt
Tellurium
Chromium....
Arsenic
Titanium
Strontium
Glucinum
Calcium
Rubidiimi
Sodium
Potassium....,
Lithium
Weisht of one
cubic fooL
pounds.
378.25 to 467.66
468
449 
435
435 ...
470
435 to 444...
442
450 ...
462
449 ..
428
418
167
160
108.5
1 165.0
1 147.0
1097.0
742.6
735.8
660.9
623.6
542.5
537.5
536.2
530.0
381.0
374.1
361.5
330.5
158.4
131.0
98.5
94.8
60.5
536
37.0
Specific Gravity.
Water = i.
6.900 to 7.500
7.50
7.20
6.969
6.970
7.530
6.977 to 7. 1 13
7.094
... 7.217
7.409
7.20
6.86
6.71
2.67
2.56
1.74
PRECIOUS STONES.
Zircon
Garnet
Malachite
Sapphire
Emerald
Do. Aqua marine..
Amethyst
Ruby
Diamond.
Specific Gravity.
.. 4.50
3.60 to 4.20
.. 4.01
3.98
.. 3.95
2.73
.. 3.92
3.95
3.50 to 3.53
Diamond, Pure
Boart
Topaz
Tourmaline
Lapis lazuli
Turquoise
Jasper, Onyx, Agate....
Beryl.:
Opal
18.68
18.40
17.60
11.91
11.80
1 0.60
10.00
8.70
8.62
8.60
8.50
6.1 1
6.00
5.80
5.30
2.54
2.10
1.58
1.52
0.97
0.86
0.59
Specific Gravity.
... 3.52
3.50
... 3.50
3.07
... 2.96
2.84
2.6 to 2.7
2.68
2.09
204
VOLUME, WEIGHT, AND SPECIFIC GRAVITY
STONES.
Specular, or red iron ore
Magnetic iron ore
Brown iron ore
Spathic iron ore ,
Clydesdale iron ores ,
Barytes
Basalt
Mica
Limestone, Magnesian
Do. Carboniferous
Marble: — Paros
African
Pyrenean ,
Egyptian, green
French
Florentine, Sienna
Trap, touchstone
Granite, Sienite, gneiss
Do. Gray
Porphyry
Alabaster, Calcareous
Do. Gypseous
Chalk, Airdried
Slate
Serpentine
Potter's Stone
Schist, Slate
Do. Rough
Lava, Vesuvian
Talc, Steatite
Rock Crystal
Quartz
Do. Crystalline
Do. for paving
Do. porous, for millstones
Do. flaky, for do
Flint
Felspar
Gypsum
Lias
Graphite
Sandstone
Tufa, volcanic
Scoria, do
Cubic feet to one
ton, solid.
cubic feet.
6.84 • *
7.05
9.16 ..
9.38
.. 11.76 ..
8.07
147 to 12.0
14.0 to 12.3
,. 12.6 ..
13.3
.. 12.7 ..
12.8
.. 13.2 ..
13.2
.. 13.2 ..
13.5
.. 13.6 ..
14.3
.. 13.2
15.2 to 12.1
12.8 to 11.8
135 to 13.1
.. 13.0 ..
15.6
14.9 to 14. 1
13.8 to 12.6
.. 12.8 ..
12.8
.. 12.8
19.9 to 12.9
21.0 to 12.8
13.3
13.6
13.8 to 13.3
13.6
.. 14.4 ..
28.5
I4.I
137
.. 13.8 ..
15.6
16.0 to 14.7
16.3
17.3 to 14.3
29,7 to 26.1
.. 43.3 •.
Weight of one cubic
foot, solid.
pounds.
••• 3274
317.6
... 244.6 ..
238.8
190.5 ..
277.5
152.8 to 187.1
160.3 to 182.7
... 178.3 ..
168.0
... I77.I ..
174.6
170.2 ..
170.2
169.6 ..
166.5
165.2 ..
157.1
169.6 ..
147.1 to 184.6
174.6 to 190.8
166.5 to 171.5
... I72.I
144.0
...150 to 159..
162.1 to 177.7
... 175.2 ..
174.6
... 174.6 ..
112.8 to 173.3
106.6 to 175.2
168.4
165.2
162.8 to 169.0
165.2
... 155.9 ..
78.6
... 159.0 ..
164.0
... I62.I ..
1434
140.3 to 152.8
137.2
129.7 to 1 57. 1
75.4 to 86.0
... 51.7 ..
Specific Gravity.
• B I • •
Water = I.
5.251
5.094
... 3.922
3.829
3.055103.380
445
2.45 to 3.00
2.57 to 2.93
2.86
2.69
. .. 2.04
2.80
... 2.73
2.73
• • i
2.67
2.65
2.52
... 2.72
2.36 to 2.96
2.80 to 3.06
2.67 to 2.75
... 2.76
2.31
2.46 to 2.55
2.60 to 2.85
... 2.81
2.80
2.80
1. 81 to 2.78
1.71 to 2.81
2.70
2.65
2.61 to 2.71
2.65
... 2.50
1.26
... 2.55
2.63
2.60
2.30
2.25 to 245
2.20
2.08 to 2.52
I.2I to 1.38
•83
1
\
OF SOLID BODIES.
201;
SUNDRY MINERAL
SUBSTANCES.'
Glass:— Flint
Green
Plate
Crown
St. Gobain
Common, with base of potash
Fine, do. do.
Common, with base of soda...
Fine, do. do. ...
Soluble
Porcelain :— China
Sevres
Portland Cement
Concrete : —
P. cement i, and shingle 10
P. cement, rubble, and sand
P. cement i, and sand 2
Roman cement i,and sand 2
Mortar.
Brick
Brickwork
Masonry, Rubble
MarL
Do. very tough
Potash
Sulphur
TUes
Rock Salt
Conunon Salt, as a solid
Clay
Sand, pure
earthy
Earth :— Potter's
Argillaceous
Light vegetable
Mud
Materials in the bed of the Clyde :
Fine sand and a few pebbles,
laid in a box, loose, not
pressed, nearly dry
Pressed
Mud at Whiteinch, dry, and ]
firmly packed, containing >
very fine sand and mica )
Wet mud, rather compact and
firm,well pressed into the box
Wet, fine, sharp gravel, well
pressed
Wet, running mud
Sharp dry sand deposit, in ^
harbour )
PortGlasgow bank (sand), wet, )
pressed into a box \
]
Cubic feet to one
ton, solid.
cubic feet.
Weight of one cubic
foot, solid.
28.7 to 23.8
.. 16.1
16.6 to 16.0
.. 17.6 ..
18.7
20.6
1 8. 1 to 16.0
20.4 to 19.5
19.4 to 15.6
22.4 to 18.9
.. 17.1
18.0
.. 18.0 ..
17. 1 to 15.9
.. 18.7 ..
18.7
.. 18.9 ..
21. 1
.. 18.9 ..
22.4
.. 25.7 ..
22.
pounds.
187.0
168.4
168.4
1559
155.3
1534
152.8
152.8
1 52. 1
77.9
148.4
1397
78 to 94
26
24
23
19
18
18.1
243
18.6
.. 139
135 to 140
127
120
109
124.7 to 135.3
..no to 115.
1 1 5.3 to 143.4
99.8 to 1 18.5
146
.. 131
1247
124.7 ••
131 to 140.7
.. II9.7 ..
1 19.7
.. II8.5 ..
106.0
.. II8.5 ..
99.8
87.3 ..
1 01. 6
Specific Gravity.
87
92
97
115
124
122X
92
120.5
Water = i.
3.00
2.70
2.70
2.50
2.49
2.46
2.45
2.45
2.44
1.25
2.38
2.24
1.25 to 1. 51
.. 2.23
2.17 to 2.25
.. 2.04
1.92
.. 1.75
2.00 to 2.17
1.76 to 1.84
1.85 to 2.30
1.60 to 1.90
2.34
2.10
2.00
2.00
2.IOOtO 2.257
1.92
1.92
1.90
1.70
1.90
1.60
1.40
1.63
... 1.39
1.48
... 1.56
... 1.95
1.99
1.97
... 1.48
... 1.93
206
VOLUME, WEIGHT, AND SPECIFIC GRAVITY
Mineral Substances {continued).
Materials in the bed of the Qyde ; —
Sand opposite Erskine House, )
wet, pressed J
Alluvial earth, pressed
Do. do. loose
Plaster: — 24 hours after using
2 months after using ...
Coal, Anthracite (see Sect. Coal)
Bituminous do. do.
Boghead (cannel) do. do.
Coke
Phosphorus
Alum
Camphor
Meltmg Ice
Cubic feet to one
ton, solid.
cubic feet.
19.3
24
.. 33
22.6
.. 25.7 ..
26.2 to 22.6
30 to 28.1
30
39 to 21.6
20.3
20.9 ..
36.3
•• 39
Weight of one cubic
foot, solid.
pounds.
116
93
67
99.2
.. 87.3 ..
85.4 to 99.1
74.8 to 81.7
74.8
57.4 to 103.5
1 10.4
.. 107.2 ..
61.7
.. 57.5 ..
Specific Gravity.
Waters X.
.. 1.86
1.49
.. 1.08
1.59
1.40
1.37 to 1.59
1.20 to 1. 31
1.20
.92 to 1.66
1.77
.. 1.72
•99
.922
• • •
COALS.
Delabeche and Playfair,)
Welsh: — ^Anthracite
Porth Mawr (highest)
Llynvi (one of the lowest)
Average of 37 samples
Newcastle: — Hedley's Hartley (highest) ...
Original Hartley (one of the lowest)
Average of 18 samples
Derbyshire and Yorkshire: — Elsecar
Butterley
Stavely
Loscoe, soft
Average of 7 samples
Lancashire: — Laffack Bushy Park (highest)
Cannel, Wigan (lowest)
Average of 28 samples
Scotch : — Grangemouth (h ighest)
Wallsend Elgin
Average of 8 samples
Irish : — Slievardagh Anthracite
Warlich's artificial fuel
Cubic feet
in a ton.
Heaped.
cubic feet.
38.4
42.0
42.0
42.7
431
45.6
453
47.4
473
44.9
48.8
47.4
42.6
46.4
45.2
40.1
41.0
42.0
35.7
32.4
Weight of one
cubic foot.
SoUd.
pounds.
85.4
86.7
80.3
82.3
81.8
78.0
78.3
80.8
79.8
79.8
79.6
79.6
84.1
76.8
794
80.5
74.8
78.6
99.6
72.2
Heaped.
pounds.
58.3
533
53.3
53.1
52.0
49.1
49.8
47.2
47.4
499
45.9
459
52.6
48.3
49.7
54.3
54.6
50.0
62.8
69.6
Specific
Gravity.
Water = i.
.37
.39
.28
•315
•31
.256
.296
.28
.27
.285
.292
•35
.23
.273
.29
.20
.259
.59
•15
OF SOLID BODIES.
207
PEAT.
{Dr, Sullivan,)
Irish peat (comprising an
average amount of water
from 20 to 25 per cent) : —
Lightest upper moss peat ...
Average lignt moss peat
Average brown peat
Compact black peat
Mean of five samples
{Another observation,)
Average upper brown peat ..
Moderately compact lower )
brown turf )
Mean of two classes
Condensed peat
{Kane and Sullivan,)
Excessively light, spongy )
surface peat )
Light surface peat
Rather dense peat
Very dense dark brown peat
Very dense blackish brown )
compact peat J
Exceedingly dense jet black \
^P«at (
Exceedingly dense, dark, \
blackish brown peat \
(/CarmarscA.)
Turfy peat, Hanover
Fibrous peat, do
Earthy peat, do
Pitchy peat, do
Cubic feet per
ton, stalked.
cubic feet.
...369.60.
254.20
...147.00.
131.28
... 99.36.
200.29
. . . I oo.O . . .
1555
...141.75.
5 1.2 to 40.0
Weight of one
cubic foot,
stalked.
pounds.
6.06
8.81
1513
17.06
22.54
11.18
.. 11.92
14.40
,. 15.80
43.75 to 56.8
Weiffht of one
cubic foot,
solid.
pounds.
62.5 to 81. 1
13.7 to 21.0
20.9 to 25.3
29.7 to 41.7
40.5 to 44.5
45.1 to 61.3
53.2 to 61.8
. . . 66.0 . . .
6.9 to 16.2
1 5.0 to 41.8
2 5.6 to 56.1
38.7 to 64.2
Specific
Gravity.
Water = i.
i.o to 1.3
.2 19 to .337
.335 to .405
.476 to .669
.65010.713
.72410.983
.72510.991
... 1.058
.II to .26
.24 to .67
.41 to .90
.62 to 1.03
FUEL IN FRANCE.
{ClaudeL)
Pure Graphite.
Anthracite
Rich coaly with ^ long flame
Dry coal, with a long flame
Rich and hard coal
Smithy coaL
Lignite
Do. bituminous
Do. imperfect
«Jayet»...r!
Bitumen, red
Do. black
Do. brown
Asphalte
Weisht of
one cubic foot
pounds.
.. 1453 ••
83.5 to 91.0
79.8 to 84.8
84.8
.. 82.3 ..
79.8 to 81. 1
77.9 to 84.2
72.3 to 74.8
68.6 to 74.2
81.7
.. 72.3 ..
66.7
... 51.7 ..
66.1
Specific Gravity.
Water = x.
.. 2.33
1.34 to 1.46
1.28 to 1.36
1.36
.. 1.32
1.28 to 1.30
1.25 to, 1.35
1. 16 to 1.20
1. 10 to 1. 19
.. 1. 16
1.07
.. 0.83
1.06
2o8
VOLUME, WEIGHT, AND SPECIFIC GRAVITY
WOODS.
Pomegranate
Boxwood
Do. of Holland
Do. of France
Lignum vitae
Ebony
Do. Green
Do. Black
Oak, Heart of.
Do. English
Do. European
Do. American, Red
Lancewood
Rosewood
Satinwood
Walnut, Green
Do. Brown
Laburnum
Hawthorn
Mulberry
Plumtree
Teak, African »
Mahogany, Spanish
Do. St. Domingo
Do. Cuba
Do. Honduras
Beech
Do. with 20 per cent, moisture.
Do. cut one year
Ash....
Weight of one
cubic foot.
Do. with 20 per cent, moisture
Acacia
Do. with 20 per cent, moisture.
Holly
Hornbeam
Yew
Birch
Elm
Do. Green
Do. with 20 per cent, moisture
YokeElm do. do
RockElm
Fir, Norway pine
Do. Red pine
Do. Spruce
Do. Larch
Do. White pine, English
Do. do. Scotch
Do. do. do. 20 per cent, moisture..
Do. Yellow pine
Do. do. American
American Pinewood, in cord (heaped)
Appletree
pounds.
.. 84.2 ..
64.8
.. 82.3 ..
56.7
40.5 to 82.9
70.5
.. 75.5 ..
74.2
.. 730 ..
58.0
43.0 to 61.7
54.2
41.8 to 63.0
64.2
599 ••
57.4
42.4 ..
574
56.7 ..
55.5
54.2 ..
Specific Gravity.
53.0 ..
46.8
349 ••
, 34.9
46.8 to 53.0
51.1
... 41.2 ..
52.4
... 43.7 ..
51.1
... 449 ••
47.5
... 47.5 ••
46.1 to 50.5
44.9 to 46. 1
343
... 47.5 ..
449
... 47.5 ..
50.0
4^* '
29.9 to 43.7
29.9 to 43.7
31.18 to 39.9
•.. 34.3 ••
34.3
30.0 ..
41.2
... 28.7 ..
21
... 45.5 ..
•• 135
1.04
1.32
0.91
.65 to 1.33
1. 13
1.21
1.19
.. 1.17
093
.69 to .99
.87
.67 to 1. 01
1.03
.. 0.96
0.92
.. 0.68
0.92
.. 0.91
0.89
.. 0.87
.98
.. 0.85
0.75
.. 0.56
0.56
0.75 to 0.85
0.82
0.66
a84
.. 0.70
0.82
.. 0.72
0.76
.. 0.76
0.74 to 0.81
0.72 to 0.74
055
.. 0.76
0.72
0.76
0.80
.. 0.74
0.48 to 0.70
0.48 to 0.70
0.50 to 0.64
.. 0.55
0.53
.. 0.49
0.66
.. 0.46
0.34
.. 0.73
OF SOLID BODIES.
209
1
Peartree....:
Orangetree %
Olivetree
Maple
Do. 20 per cent, moisture
Servicetree.
Cypress, cut one year
Planetree
Vine tree
Aspentree.,
.\ldertree
Do. 20 per cent, moisture
Sycamore
Cedar of Lebanon
Bamboo
Poplar.
Do. White
Do. 20 per cent, moisture
Willow
Cork.
Elder pith
INDIAN WOODS.
{Berkley^
Northern Teak
Southern Teak
Jungle Teak
Blackwood
Khair
Enroul
Red Eyne
Bibla
Peon
Kullum
Hedoo.
COLONIAL WOODS.
Jamaica:—
Black heart ebony
Lignum vitae
Small leaf.
Neesberry bullettree
Red bullytree
Iron wood
Sweet wood
Fustic
Satin candlewood ,
Bastard cabbage bark
White dogwood
Black do
Gynip
Weiffht of one
cubic fooL
pounds.
455
443 ••
42.4
40.5 ..
41.8
41.8 ..
41.2
40.5 ..
374
374 ..
34.9
37.4 ..
36.8
30.6 to 35.5
19.5 to 24.9
.. 24.3 ..
20.0 to 31.8
29.9 ..
30.6
15.0 ..
4.74
55
48
41
56
11
63
68
56
39
41
39
40.5
74.2 ..
to 73.0
73.0 ..
65.5
62.36 ..
61.7
60.5 ..
60.5
599 ••
58.6
58.6 ..
58.0
58.0 ..
Specific Gravity.
073
.. 0.71
0.68
.. 0.65
0.67
.. 0.67
0.66
.. 0.65
0.60
0.60
0.56
0.60
0.59
0.49 to 0.57
0.31 to 0.40
•• 0.39
0.32 to 0.51
.. 0.48
0.49
.. 0.24
0.076
0.882
0.770
0.658
0.898
1. 171
I.0I4
I.09I
0.898
0.625
0.658
0.625
1. 19
0.65 to 1. 17
.. 1. 17
1.05
I.OO
0.99
.. 0.97
0.97
.. 0.96
0.94
.. 0.94
093
.. 0.93
14
210
VOLUME, WEIGHT, AND SPECIFIC GRAVITY
Colonial Woods {continual).
Jamaica {continued): —
Wild mahogany
Cashaw
Wild orange
Sweet do
Bullettree (bastard)
Tamarind
Do. wild
Prune
Yellow Sanders
Beech
French Oak
Broad Leaf
Fiddle Wood
Prickle Yellow
Boxwood
Locusttree
Lancewood
Green Mahogany
Yacca.
Cedar
Calabash
Bitter Wood
Blue Mahoe
Average of 36 woods of Jamaica
New South Wales:—
Box of Ilwarra
Do. Bastard
Do. True, of Camden
Mountain Ash
Kakaralli
Iron Bark
Do. broadleaved
Woolly Butt
Black Do
Water Gum
Blue Do
Cog Wood
Mahogany
Do. swamp
Gray Gum
Stringy Bark
Hickory
Forest Swamp Oak
Mean of 18 woods of New South Wales..
British Guiana :
Sipiri, or Greenhcart
Wallaba
Brown Ebony
Letter Wood
Cuamara or Tonka
Monkey Pot
Mora
Weiffht of one
cubic foot.
pounds.
.. 57.4 ..
574
53.0 to 56.7
493
.. 56.1 ..
54.2
46.8
. 53.6
.. 536 ..
52.4
48.0
.. 44.3 ..
430
.. 43.0 ..
42.4
.. 42*4 *•
41.2
.. 393 .•
36.2
.. 349 ••
34.3
.. 337 ..
52.1
730
69.8
60.5
69.2
68.6
64.2
63.6
63.0
55.5
63.6
52.4
599
59.2
53.6
58.0
53.6
46.8
41.2
599
65.5 to 68.0
64.8
64*2
62.36
.. 61.7 ..
58.6
57.4
Spedfic Gtavity.
0.92
0.92
.85 to 0.91
0.79
0.90
0.87
0.75
0.86
0.86
0.84
0.77
0.77
0.71
0.69
0.69
0.68
0.68
0.66
0.63
0.58
0.56
0.55
0,54
0.835
. I.I7
1. 12
0.97
I.II
1. 10
1.03
1.02
l.OI
0.89
1. 00
0.84
0.96
0.95
0.86
0.93
a86
0.75
0.66
0.96
1.05 to 1.09
1.04
1.03
1. 00
.. 0.99
0.94
.. 0.92
J
OF SOLID BODIES.
211
Colonial Woods {continued),
British Guiana {continued) :—
Ducaballi
Cabacalli
Kaiecriballi
Sirabuliballi
Buhuradda
Buckati
Houbaballi
Baracara.
White Cedar
Locusttree.
Cartan
Purple Heart
Bartaballi
Crabwood :
SilverbaJli
Mean of 22 woods of British Guiana.
' WiUow
Oak
WOODCHARCOAL (as powder).
{Ciaudel)
Alder.
Limetree
Poplar
Average of 5 charcoals,
WOODCHARCOAL (in small pieces, heaped).
Walnut
Ash
Beech
Vokeelm.
Appletree
White Oak.....
^errytree....
Birch.
EJm
Yellow Pine..
Chestnuttree.
Poplar ,
Cedar
{ClaudeL)
Average of 13 charcoals
Gunpowder
WOODCHARCOAL (as made, heaped).
Oak and Beech..
Birch
Pine.
Average.
Weight of one
cubic foot.
pounds.
56.7
55.5
54.2
52.4
50.5
50.5
50.5
50.5
48.0
44.3
437
42.4
399
37.4
343
46.1
96.7
954
92.9
91.0
90.4
93.5
393
343
32.5
28.7
28.7
26.2
25.6
22.5
22.5
20.6
17.5 ...
15.6
15.0 ...
25.3
i09.itoii4.7
15 to 15.6
137 to 14.3
12.5 to 13.1
14
Specific Gravity.
0.91
0.89
0.87
0.84
0.81
0.81
0.81
0.81
0.77
0.71
0.70
0.68
0.64
0.60
0.55
0.74
155
1.53
1.49
1.46
145
1.50
.. 0.63
0.55
.. 0.52
0.46
.. 0.46
0.42
.. 0.41
0.36
.. 0.36
.. 0.28
0.25
.. 0.24
0.405
1.75 to 1.84
0.24 to 0.25
0.22 to 0.23
0.20 to 0.21
,.. 0.225
212
WEIGHT AND VOLUME OF
ANIMAL SUBSTANCES.
{Claudel)
Pearls
Coral
Ivory
Bone
Wool
Tendon
Cartilage
Crystalline humour
Human body
Nerve
Wax
White of whalebone
Butter
Pork fat
Mutton fat
Animal charcoal, in heaps
VEGETABLE SUBSTANCES.
{Claudel)
Cotton
Flax
Starch
Fecula
Gum — Myrrh
Do. Dragon
Do. Dragon's blood
Do. Sandarac
Do. Mastic
Resin — Jalap
Do. Guayacum
Do. Benzoin
Do. Colophany
Amber, Opaque
Do. Transparent
Guttapercha
Caoutchouc
Grain, Wheat, heaped
Do. Barley, do
Do. Oats, do
Weight of one
cubic foot.
pounds.
... 169.6 ...
167.7
... 1 19.7 ...
1 1 2.2 to 124.7
... 100.4 ...
69.8
68.0 ...
66.7
64,9
... 59.9 ...
58.7
... 58.7 ...
58.7
574
50 to 52
121.6
1 1 1.6
954
935
84.8
82.3
74.8
68.0
66.7
76.1
748
68.0
66.7
68.0
67.3
60.5
58.0
46.7
36.6
31.2
Specific Gravity.
.. 2.72
2.69
1.92
1.80 to 2.00
I.61
1. 12
1.09
1.08
.1.07
1.04
0.96
0.94
0.94
0.94
0.92
0.80 to 0.83
O,
o,
o,
o
o
95
79
53
50
36
32
20
09
07
22
20
09
07
09
08
97
93
75
59
50
VARIOUS SUBSTANCES.
213
TABLE No. 66.— WEIGHT AND VOLUME OF VARIOUS
SUBSTANCES. {Tredgold,)
SUBSTANCE.
Lead (cast in pigs)
Iron (cast in pigs)
Limestone or marble (in blocks)
Granite (Aberdeen, in blocks) ...
Granite (Cornish, in blocks)
Sandstone (in blocks)
Portland stone (in blocks)
Potter's clay
Loam or strong soil
Bath stone (in blocks)
GraveL
Sand.
Bricks (common stocks, dry)
Culm
Water (river)
Splint coal
Oak (seasoned)
Coal (Newcastle caking)
Wheat
Barley
Red fir
Hay (compact, old)
Cubic feet per
ton, in bulK.
cubic feet.
4
6.25
13
13.5
14
16
17
17
18
18
21
23.5
24
36
36
395
43
45
47
59
59
280
Weight of one
cubic toot, in bulk,
lbs.
567
360
172
166
164
141
132
130
126
123.5
109
95
93
63
62.5
57
52
50
48
38
38
8
TABLE No. 67.— WEIGHT AND VOLUME OF GOODS CARRIED ON
THE BOMBAY, BARODA, AND CENTRAL INDIA RAILWAY.
By Colonel J. P. Kennedy, Consulting Engineer of the Railway.
No. of
kbuL
I
2
3
4
I
7
Class I.
CLASSIFICATION OF GOODS CONVEYED.
Unpressed cotton . . . .
Furniture
Halfpressed cotton..
Cotton seeds
Wool
Fruit and vegetables.
Eggs....
Averages
Cubic feet
per ton.
Weight per !
cubic foot.
cubic feet
... 224 ...
200
... 186 ...
186"
... 140 ...
100
... 90...
lbs.
... 10 ...
II
... 12 ...
12
... 16 ...
22
... 25 ...
...174...
... 13 ...
Cubic feet
per ton,
in bulk
(estimated).
cubic feet.
.. 280
250
..233
,..175
125
... 113
... 217
214
WEIGHT AND VOLUME OF GOODS.
Goods conveyed over the Indian Railway {continued).
No. of
kind.
8
9
lO
II
12
M
17
i8
19
20
21
22
Class 2.
classification of goods conveyed.
23 ...
24
25 ...
26
27 ...
28
29 ...
30
31 .
32
33 
34
35 ..•
36
Class 3.
37 ..•
38
39 ••
40
41 ...
42
43 ...
Qass 4.
Grass
Sundries
Bagging .,.,
Commissariat stores
Fullpressed cotton
Flax and hemp
Groceries
Grains and seed
Twist ^
Sugar
Soap
Firewood
Salt
Lime
Dry Fruits
Averages
Jagree (Molasses)
Kupas (Seed cotton)
Mowra (flowers which produce spirit)
Timber
Ghee (clarified butter)
Oil
Piece goods
Rape
Beer and Spirits
Coal
Paper
Tobacco
Opium
Machinery
Averages
Cutlery
Potash
Sand
Colour
Bricks
Stone
Metal
Averages
Averages of all classes
Cubic feet
per ton.
culnc feet.
80
80
70
70
70
70
60
60
60
56
5>
51
50
Weight per
cubic foot.
60
45
45
45
45
40
40
40
40
36
28
28
28
26
25
... 41
20
20
20
18
17
15
5
... II
. . . 64.4 '
lbs.
28
28
32
32
32
32
37
37
37
40
40
40
44
44
45
37
50
50
50
^A
5^
5^
5^
56
62
80
80
80
86
90
54
112
112
112
124
132
148
443
354
Cubic feet
per ton,
m bulk
(estimated).
203 ...
cubic feet.
...ICX>
ICO
... 87
87
... 87
87
... 75
75
... 75
70
... 70
70
... 64
64
... 63
... 75
56
56
50
50
50
50
45
35
35
35
33
31
51
25
25
20
22
21
19
6X
... 14
80
J^oU, — The last column has been added bv the author; the quantities are calculated by
adding onefourth to the quantities in the third column, to give approximate estimate of
the volume occupied in waggons by the goods, or the space required to load a ton of each
kind. Sand, No. 39, lies solid in any situation.
WEIGHT AND SPECIFIC GRAVITY OF LIQUIDS.
215
TABLE No. 68.WEIGHT AND SPECIFIC GRAVITY OF
LIQUIDS.
LIQUIDS AT 33* F.
Mercury
Bromine.
Sulphuric acid, maximum concentration..
Nitrous acid
Chloroform.
Water of the Dead Sea
Nitric acid, of commerce
Acetic acid, maximum concentration
Milk.
Sea water, ordinary
Pure water (distilled) at 39°.! F
Wine of Bordeaux.
Do. Burgundy
Oil, lintseed
j Do. poppy
1 Do. rapeseed
Do. whale
Do. olive
Do. turpentine '.
Do. potato
Petroleum
Naphtha
Ether, nitric
Do. sulphurous
Do. nitrous
Do. acetic
Do. hydrochloric
Do. sulphuric
Alcohol, proof spirit
Do. pure
Benzine
Wood spirit
Weight of one
cubic foot.
pounds.
848.7
185.I
1 14.9
96.8
95.5
774
76.2
67.4
64.3
64.05
62.425
62.1
61.9
58.7
58.1
57.4
57.4
57.1
54.3
51.2
54.9
69.3
67.4
55.6
55.6
54.3
44.9
57.4
493
53.1
499
Weight of
one gallon.
pounds.
136.0 .
29.7
18.4 ,
15.5
15.3 .
12.4
12.2 .
10.8
10.3 ,
10.3
1 0.0 .
99
99 •
94
93 .
9.2
9.2 .
8.7 .
8.2
8.8 .
8.5
II. I ,
10.8
8.9 .
8.9
8.7 ,
7.2
, 9.2
7.9
, 8.5
8.0
Specific
Gravity.
Water = i.
..13.596
2.966
.. 1.84
1.55
.. 1.53
1.24
.. 1.22
1.08
.. 1.03
1.026
.. I.OOO
0.994
.. 0.991
0.94
.. 093
0.92
.. 0.92
0.915
.. 0.87
0.82
.. 0.88
0.85
.. I. II
1.08
.. 0.89
0.89
.. 0.87
0.72
.. 0.92
0.79
.. 0.85
0.80
2l6
WEIGHT, ETC., OF GASES AND VAPOURS.
TABLE No. 69.— WEIGHT AND SPECIFIC GRAVITY OF
GASES AND VAPOURS.
GASES AT 32* F. AND UKDBR ONE
ATMOSPHERE OP PRESSURE.
Vapour of mercury (ideal)
Vapour of bromine
Chloroform
Vapour of turpentine
Acetic ether
Vapour of benzine
Vapour of sulphuric ether ....
Vapour of ether (.'*)
Chlorine
Sulphurous acid
Alcohol
Carbonic acid (actual)
Do. (ideal)
Oxygen
Air
Nitrogen
Carbonic oxide
Olefiant gas
Gaseous steam
Ammoniacal gas
Light carburetted hydrogen ..
Coalgas (page 458)
Hydrogen
Volume of one
pound weight.
cubic feet.
.. 1.776...
2.236
.. 2.337...
2.637
.. 4*075 •••
4.598
.. 4.790 ...
4.777
.. 5.077 ...
5.513
. . 7.679 . . .
8.IOI
.. 8.160 ...
r 1.205
.. 12.307 •••
12.727
.. 12.004 •••
12.580
• .19.913 ...
21.017
..22.412 ...
28.279
179.00 ...
Weight of one cubic foot.
in pounds.
...0.563 ..
0.447
...0.428 ...
0.378
...0.245 "
0.217
...0.209 ...
0.206
...0.197 ...
O.1814
...ai302 ...
0.12344
...0.12259
0.089253
...0.080728
0.078596
...0.0781 ...
0.07808
.0.05022
0.04758
...0.04462
0.03536
...0.005594
m ounces.
..9.008 .
7.156
.6.846 .
6.042
.3927 .
3.480
.3.340 .
3.302
.3.152 .
2.902
...2.083 •
1.975
.. .1.961 .
1.428
.. .1.29165
1.258
...1.250 .
1.249
...0.8035 •
0.7613
...0.7139..
0.5658
...0.0895
Specific
Gravity.
Air = X.
.. 6.9740
5.5400
.. 5.3000
4.6978
.. 3.0400
2.6943
.. 2.5860
2.5563
.. 2.4400
2.2470
.. I.6130
1.5290
.. 1. 5180
1. 1056
.. I.OOOO
0.9736
.,0.9674.
0.9672
..0.6220
0.5894.
.0.5527
0.4381
..0.0692
TABLES OF THE WEIGHT OF IRON AND
OTHER METALS.
Wrought Iron. — According to Table No. 65 of the Weight and Specific
Gravity of Solids, the weight of a cubic foot of wrought iron varies, for
various qualities, from 466 pounds to 487 pounds per cubic foot, and the
average weight, taken for purposes of general calculation, is 480 pounds per
cubic foot. This average weight is equivalent to a weight of 40 pounds per
square foot, i inch in thickness — a convenient unit, which is usually
employed in the development of tables of weights of iron for engineering
and manufacturing purposes. The extremes of variation from this medium
unit, extend from ^^ pound less, to about fi pound more than 40 pounds
per square foot, or from 2.2 to 1.5 per cent, either way — a deviation, the
extent of which is of little or no practical consequence, and which, at all
events, is comprehended in the percentages allowed in the framing of
estimates.
The average weight of a cubic inch of ^vrought iron is
i?^ =.277 pound,
1720
or onetenth more than a quarter of a pound. For a round number, when
cubic inches are dealt with, it may be, and is usually, taken as .28 pound,
which is only fourfifths of i per cent, more than the medium weight, and
corresponds to a weight of 483.84 pounds per cubic foot, or to 40.32
pounds per square foot, i inch thick, or to 10 pounds per lineal yard,
I inch square.
The volume of i pound of wTought iron is 3.6 cubic inches.
Sied. — ^The weight of a cubic foot of steel varies from 435 pounds to
493 pounds per cubic foot, and the average weight is about 490 pounds
per cubic foot. For convenience of calculation, the average weight is taken
in the following tables, as 489.6 pounds per cubic foot, for which the
specific weight is 1.02, when that of wrought iron = 1.00. The weight of a
square foot, i inch thick, is 40.8 pounds; of a lineal yard, 10.2 pounds;
and of a cubic inch, .283 pound.
The volume of i pound of steel is 3.53 cubic inches.
Cast Iron, — The weight of a cubic foot of cast iron varies from 378^^
pounds to 467^ pounds per cubic foot, and the average weight is taken as
450 ix)unds. The weight of a square foot, i inch thick is, therefore, 37.5
pounds; of a lineal yard, i inch square, 9.375 pounds; and a cubic inch,
.26 pound. The specific weight is .9375.
The volume of i pound of cast iron is 3.84 cubic inches.
The following data, for the weight of iron, are abstracted for readiness
of reference: —
2l8 WEIGHT OF METALS.
Wrought Iron, Rolled.
I cubic foot, 480 pounds, or 4.29 cwts.
I square foot, i inch thick, 40 pounds.
I square foot, 3 inches thick, 120 pounds, or 1.07 cwts.
. 3 square feet, i inch thick, 120 pounds, or 1.07 cwts.
I lineal foot, i inch square, 3 ^ pounds, or .03 cwt
I cubic inch, say 0.28 pound.
3.6 cubic inch, i pound.
I lineal yard, i inch square, 10 pounds.
I lineal foot, 3 inches square, 30 pounds.
I lineal foot, 6 inches square, 120 pounds, or 1.07 cwts.
I lineal foot, 3 inches by i inch thick, 10 pounds.
I lineal foot, ]/i inch in diameter,.... 2 pounds.
1 lineal foot, 2 inches in diameter,... 10.5 pounds.
I lineal foot, 6 J^ in. in diameter, about i cwt
Cast Iron.
I cubic foot, 450 pounds, or 4 cwts.
5 cubic feet, i ton.
I square foot, i inch thick, 37.5 pounds.
I squarefoot, 3 inches thick (^ cub. ft), 112.5 pounds, or i cwt
3 square feet, i inch thick, 112.5 pounds, or i cwt
I cubic inch, 0.26 pound.
3.84 cubic inches, i pound.
The Table No. 70 contains the weight of iron and other metals for the
following volumes : —
I cubic foot
I square foot, i inch thick, or ^ji^ih. of a cubic foot
I lineal foot, i inch square, or 7xath of a square foot
I cubic inch, or Viath of a lineal foot.
A sphere, i foot in diameter.
The specific gravity due to the respective weights per cubic foot is also
given, and likewise the specific weight or heaviness, taking the weight of
wrought iron as i, or unity.
The next Table, No. 71, contains the volumes of iron and other metals
for the following weights : —
I ton, in cubic feet
I cwt, in square feet, i inch thick.
I cwt, in lineal feet, i inch square.
I pound, in cubic inches.
I ton, as a sphere, in feet of diameter.
I ton, as a cube, in feet of lineal dimension.
The next Table, No. 72, contains the weight of 1 square foot of metals of
various thickness, advancing by sixteenths and by twentieths of an inch, up
to I inch in thickness.
The fourth Table, No. 73, contains the weight of prisms or bars of iron,
and other metals, or metals of any other uniform section, for given sectional
areas, varying from .1 square inch to 10 square inches of section, advancing
by onetenth of an inch, for i foot and i yard in length.
TABLES Ot WEIGHT AND VOLUME OF METALS.
219
This table is useful in calculations of the weights of bars of every form,
rails, joists, beams, girders, tubes, or pipes, &c., when the sectional area
is given.
The table is available for finding the weight of a metal for any sectional
area up to 100 square inches, by simply advancing the decimal points one
place to the right; or, in round numbers, up to 1000 square inches, by
advancing the decimal points two places. For example, to find the weight
of wrought iron having a sectional area of 17 square inches:—
For 1.7 square inches, the weight per foot is 5.67 pounds.
For 17 square inches, the weight per foot is 56.7 pounds.
For 170 square inches, the weight per foot is 567 pounds.
Table No. 70. — Weight of Metals.
Mbtau
Wrought Iron.
Cast Iron
1 Steel
j Copper, Sheet
Copper, Hammered
Tin
Zinc
Lead.
Brass, Cast.
Brass, Wire,
Gon Metal
Silver
Gold
Platinum
Cubic Foot.
lbs. or cwts.
480 or
450 or
'489.6 or
549 or
556 or
462 or
437 or
712 or
505 or
533 or
524 or
655 or
1 200 or
1342 or
4.29
4.02
437
4.90
4.96
4.13
3.90
6.36
4.51
4.76
4.68
5.85
10.72
12.00
Square Foot,
X inch Thick.
lbs. or cwts.
40 or
37.5 or
40. 8 or
45.8 or
46.3 or
38.5 or
36.4 or
59. 3 or
42. 1 or
44.4 or
43 7 or
54. 6 or
loo.oor
iii.8or
357
335
364
409
413
344
325
530
375
396
390
488
893
1. 000
Lineal
Foot,
I Inch
Square.
lbs.
3.333
3.125
3.400
3813
3.861
3.208
3.035
4.944
3.507
3.701
3.639
4.549
8.333
9.320
Cubic
Inch.
lb.
.278
.260
.283
.318
.322
.268
•253
.412
.292
.308
■304
.379
.694
.777
Sphere,
I Foot
Dia
meter.'
lbs.
251
236
257
287
291
242
229
373
264
279
274
343
628
703
Specific
Gravity.
Water
= x.
7.698
7.217
8.805
8.917
7.409
7.008
II. 418
8.099
8.548
8.404
10. 505
19.245
21.522
Specific
eight.
apei
Wei
Wro'ght
Iron=z.
1.000
.9375
1.020
1. 144
1. 158
.962
.910
1.483
1.052
I.IIO
1.092
1.365
2.500
2.796
Table No. 71. — ^Volume of Metals for given Weights.
Metal.
Wromgfht Iron
Cast &on
Steel
Copper, Sheet ....
Copper, Hammered
Tm
Zinc
Lead
Brass, Cast..
Brass, Wire.
Gun Metal
Silver
Gold
Platinum
Cubic Feet
to a Ton.
cubic feet
4.67
4.98
4.58
4.08
4.03
4.86
5.13
3.15
4.44
4.20
4.28
3.42
1.87
1.67
Square Feet,
X Inch Thick,
toacwt.
square feet.
2.80
2.99
2.75
2.44
2.42
2.91
3.08
1.89
2.67
2.30
2.56
2.05
1. 12
I.OO
Lineal Feet,
I In. Square,
to a cwt.
feet.
33.6
35.8
32.9
29.4
29.0
34.9
36.8
22.7
31.9
30.3
30.8
24.6
13.4
12.0
Cubiclnches
to a lb.
cubic inches.
3.60
3.84
3.53
3.15
3."
374
395
2.43
3.42
3.24
330
2.64
1.44
1.29
Diameter
of a Sphere
of I Ton.
feet.
2.07
2.12
2.26
1.98
1.98
2.10
2.14
1. 81
2.04
2.00
2.02
1.87
1.59
1.47
Side of a
Cube of
z Ton.
feet.
1.67
I.71
1.66
1.60
159
1.69
1.73
1.47
1.64
1. 61
1.62
1.51
1.28
1.19
220
WEIGHT OF METALS.
Table No. 72. — Weight of i Square Foot of Metals.
Thickness advancing by Sixteenths of an Inch.
Thick
ness.
inch.
3/16
%
s/16
H
7/16
9/16
H
"As
H
'3/16
H
15/16
I
Wro't
Ikon.
Specific
wt. =1,
lbs.
2.50
5.00
7.50
10.
12.5
150
17.5
20.0
22.5
25.0
27.5
30.0
32.5
35.0
37.5
40.0
Cast
Iron.
Specific
wt.=.937S.
lbs.
2.34
4.69
7.0^
9.38
II.7
14. 1
16.4
18.7
21. 1
23.5
25.8
28.1
30.5
32.8
352
37.5
Stbel.
Specific
Wt.= 1.02.
lbs.
2.55
5.10
7.65
10.2
12.8
»53
17.9
20.4
23.0
28.1
30.6
33.2
357
38.3
40.8
Copper.
Specific
wt.=i,i6.
lbs.
2.89
579
8.68
II. 6
14.5
17.4
20.3
23.2
26.0
28.9
31.8
34.7
37.6
40.5
43.4
46.3
Tin.
Specific
wt=.962.
lbs.
2.41
4.81
7.22
963
12.0
14.4
16.8
19.3
21.7
24.1
26.5
28.9
313
337
36.1
38.5
Zinc.
Specific
wt.=.9io.
lbs.
2.28
4.55
6.83
9.10
II.4
137
15.9
18.2
20.5
22.8
25.0
27.3
29.6
319
34.1
36.4
Brass.
Specific
wt. =1.052.
lbs.
2.63
5.26
7.89
10.5
13.2
il8
18.4
21. 1
237
26.3
28.9
31.6
34.2
36.8
395
42.1
Gun
Metal.
Specific
wt.= 1.092.
lbs.
2.73
5.46
8.19
10.9
137
16.4
19. 1
21.9
24.6
27.3
30.0
32.8
35.0
38.2
41.0
437
Lead.
Specific
wt.=i.48.
lbs.
3.71
7.41
II. I
14.8
18.5
22.2
25.9
29.7
334
37. i
40.8
44.5
48.2
519
556
593
Thickness advancing by Twentieths of
an Inch.
inch.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
.05
2.00
1.88
2.04
2.32
'•§3
1.82
2. II
2.19
2.96
.10
4.00
3.75
4.08
4.63
3.85
364
4.21
4.37
5.93
.15
6.00
5.63
6.12
6.95
5.78
5.46
6.32
6.56
8.90
.20
8.00
7.50
3.16
9.26
7.70
7.28
8.42
8.74
"•§
.25
10.0
9.38
10.2
II. 6
9.63
9.10
10.5
10.9
14^
.30
12.0
"3
12.2
13.9
II. 6
10.9
12.6
131
17.8
.35
14.0
131
14.3
16.2
135
12.7
147
15.3
20.8
.40
16.0
15.0
16.3
18.5
154
14.6
16.8
17.5
23.7
.45
18.0
16.9
18.4
20.8
17.3
16.4
18.9
19.7
26.7
.50
20.0
18.8
20.4
23.2
193
18.2
21. 1
21.9
29.7
.55
22.0
20.6
22.4
25S
21.2
20.0
23.2
24.0
327
.60
24.0
22.5
245
27.8
23.1
21.8
25.3
26.2
35.6
.65
26.0
24.4
26.5
30.1
25.0
237
27.4
28.4
38.6
.70
28.0
26.3
28.6
32.4
27.0
255
29.5
30.6
41.5
•75
30.0
28.1
30.6
34.7
28.9
27.3
31.6
32.8
445
.80
32.0
30.0
32.6
37.0
30.8
29.1
337
35.0
47.5
.85
34.0
319
34.7
39.4
32.7
30.9
35.8
37.2
50.4
.90
36.0
33.8
^tl
41.7
34.7
32.8
379
39.3
534
.95
38.0
356
38.8
44.0
36.6
346
40.0
41.5
56.3
I.OO
40.0
37.5
40.8
46.3
38.5
36.4
42.1
437
59.3
Note to Table *j% next page. — To find the weight of I lineal foot or I lineal yard of
hammered iron, copper, tin, zinc, or lead, multiply the tabular weight for rolled wrought
iron of the given dimensions by the following multipliers, respectively : —
Exact. Approximate.
Hammered Iron 1.008 i.oi equivalent to I percent more.
Copper 1. 158 1. 16 ,, 16 ,, more.
Tin 962 96 ,, 4 ,, less.
Zinc 91 91 ,, 9 „ less.
Lead I'483 1.48 ,, 48 „ more.
WEIGHT OF METALS OF A GIVEN SECTIONAL AREA. 221
Table No. 73. — Weight of Metals, of a given Sectional Area,
PER Lineal Foot and per Lineal Yard.
Rolled
Wrought Iron.
Cast Iron.
Stesl.
Brass.
Gun Mbtal.
Sect.
' Akea
Sp. Wcight=x.
Sp.Weight=.9375.
Sp. Weight=i.o2.
Sp.Weight=i.o52.
Sp» Weight= 1.092.
1
I Foot.
I Yard.
X Foot.
I Yard.
I Foot.
I Yard.
I Foot
1 Yard.
I Foot.*
I Yard.
, sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
.1
333
I.OO
•313
.938
.340
1.02
•351
1.05
■364
1.09
.2
.667
2.00
.625
1.88
.680
2.04
.701
2.10
.728
2.18
•3
I.OO
3.00
.935
2.81
1.02
3.06
1.05
3.16
1.09
3.28
.4
133
4.00
1.25
3.75
1.36
4.08
1.43
4.21
1.46
437
•5
1.67
5.00
1.56
4.69
1.70
5 10
1.75
5.26
1.82
5.46
.6
2.00
6.00
1.88
5.63
2.04
6.12
2.II
6.31
2.18
6.55
.7
233
7.00
2.19
6.56
2.38
7.14
2.46
736
2.55
7.64
.8
2.67
8.00
2.50
7.50
2.72
8.16
2.81
8.42
2.91
8.74
•9
3.00
9.00
2.81
8.44
3.06
9.18
3.16
947
3.28
9.83
1 1.0
3.33
10.
3.15
9.38
3.40
10.2
3.51
I0.5
364
10.9
I.I
367
II.O
344
10.3
374
II. 2
3.86
II. 6
4.00
12.0
1.2
4.00
12.0
375
"3
4.08
12.2
4.21
12.6
437
13.1
13
4.33
130
4.06
12.2
4.42
133
4,56
137
4.73
14.2
1.4
4.67
14.0
4.38
13 1.
4.76
14.3
4.91
14.7
5.10
15.3
iS
5.00
15.0
4.69
14. 1
S'lo
153
5.26
15.8
5.46
16.4
1.6
5.33
5.67
16.0
5.00
15.0
544
16.3
5.61
16.8
5.82
>75
17
17.0
5.31
159
5.78
17.3
5.96
17.9
6.19
18.6
1.8
6.00
18.0
563
16.9
6.12
18.4
6.31
18.9
6.55
197
19
6.33
19.0
594
17.8
6.46
19.4
6.66
20.0
6.92
20.8
2.0
6.67
20.0
6.25
18.8
6.80
20.4
7.01
21.0
7.28
21.8
2.1
7.00
21.0
6.56
19.7
7.14
21.4
736
22.1
7.64
22.9
2.2
7.33
22.0
6.88
20.6
7.48
22.4
7.72
23.1
8.01
24.0
23
7.67
23.0
7.19
21.6
7.82
235
8.07
24.2
8.37
25.1
2.4
8.00
24.0
7.50
22.5
8.16
24.5
8.42
253
8.74
26.2
2.5
f.33
25.0
7.81
234
8.50
255
8.77
26.3
9.10
27.3
2.6
867
26.0
8.13
24.4
8.84
26.5
9.12
27.4
9.46
28.4
H
9.00
27.0
8.44
253
9.18
27.5
9.47
28.4
9.83
29.5
2.8
9.33
28.0
8.75
26.3
9.52
28.6
9.82
29.5
10.2
30.6
2.9
9.67
29.0
9.06
27.2
9.86
29.6
10.2
30s
31.6
10.6
^H
3.0
lO.O
30.0
9.38
28.1
10.2
30.6
10.5
10.9
32.8
31
10.3
310
9.69
29.1
10.5
31.6
10.9
32.6
"3
339
32
10.7
32.0
10.0
30.0
10.9
32.6
II. 2
337
11.7
34.9
33
II.O
330
10.3
309
II. 2
337
I1.6
34.7
12.0
36.0
34
"•3
34.0
10.6
319
11.6
34.7
11.9
35.8
12.4
37.1
3.5
1 1.7
35.0
10.9
32.8
11.9
357
12.3
36.8
12.7
38.2
3.6
12.0
36.0
"3
33.8
12.2
36.7
12.6
37.9
13.1
39.3
3.7
*2.3
37.0
11.6
34.7
12.6
37.7
130
38.9
13.5
40.4
3.8
12,7
380
11.9
35.6
12.9
38.8
133
40.0
13.8
41.5
3.9
13.0
39.0
12.2
36.6
133
398
13.7
41.0
14.2
42.6
4.0
»3.3
40.0
12.5
37.S
13.6
40.8
14.0
42.1
14.6
43.7
4.1
13.7
41.0
12.8
38.4
139
41.8
14.4
431
14.9
44.8
42
14.0
42.0
131
394
14.3
42.8
14.7
44.2
15.3
45.9
4.3
H3
43.0
^H
40.3
14.6
43.9
15. 1
45.2
15.7
46.9
44
14.7
44.0
138
41.3
15.0
44.9
^H
46.3
16.0
48.0
45
15.0
45.0
14. 1
42.2
15.3
459
15.8
4Z3
16.4
49.1
4.6
15.3
46.0
14.4
431
15.6
46.9
16. 1
48.4
16.7
50.2
4.7
15.7
47.0
14.7
44.1
16.0
47.9
16.5
49.4
17. 1
51.3
4.8
16.0
48.0
15.0
45.0
16.3
49.0
16.8
50.5
17.5
52.4
4.9
16.3
49.0
153
45.9
16.7
50.0
17.2
51.6
17.8
53.5
5.0
16.7
50.0
15.6
46.9
17.0
51.0
17.5
52.6
18.2
54.6
222
WEIGHT OF METALS.
Table No 73 {continued).
R0M.RD
Wrought Irom.
Cast Iron.
Stbel.
Brass.
Gun Metal.
Sect.
Sp. Weight=x.
Sp.Weight — 9375
Sp.Weight=z.03.
Sp.Weight=x.o53.
Sp.Weight=x.o9a.
Akra
mmW^M»*\»
xFoot
xYard.
I Foot
xYard.
I Foot
xYard.
I Foot
xVard.
X Foot
I Yard.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
5.1
17.0
51.0
15.9
47.8
17.3
52.0
17.9
537
18.6
m
5.2
17.3
52.0
16.3
48.8
17.7
53
18.2
54 Z
18.9
53
17.7
53.0
16.6
49.7
18.0
54.1
18.6
55.8
X93
579
5.4
18.0
54.0
16.9
50.6
18.4
55.1
18.9
56.8
19.7
58.9
55
18.3
55.0
17.2
51.6
18.7
56.1
19.3
57.9
58.9
20.0
60.0
5.6
18.7
56.0
17.5
.52.5
19.0
57^
19.6
20.4
20.8
61. 1
5.7
19.0
57.0
17.8
53.4
19.4
58.1
20.0
60.0
62.2
5.8
19.3
58.0
18. 1
544
19.7
59.2
20.3
61.0
21.1
633
59
19.7
59.0
18.4
55.3
20.1
60.2
20.7
62.1
21.5
21.8
64.4
6.0
20.0
60.0
18.8
56.3
20.4
61.2
21.0
63.1
65.5
6.1
20.3
61.0
19. 1
57.2
20.7
62.2
21.4
64.2
22.2
66.6
6.2
20.7
62.0
19.4
58.1
21. 1
63.2
21.7
65.2
22.6
67.7
6.3
21.0
63.0
197
59.1
21.4
64.3
22.1
66.3
22.9
68.8
6.4
21.3
64.0
20.0
60.0
21.8
65.3
22.4
67.3
23.3
69.9
6.5
21.7
65.0
20.3
60.9
22.1
66.3
22.8
68.4
23.7
70.9
6.6
22.0
66.0
20.6
61.9
22.4
67.3
23.1
69.4
24.0
72.0
H
22.3
67.0
20.9
62.8
22.8
68.3
23.5
70.5
24.4
24.8
731
6.8
22.7
68.0
21.3
63.8
23.1
69.4
23.9
71.5
74.2
6.9
23.0
69.0
21.6
64.7
23.5
70.4
24.2
72.6
25.1
753
7.0
23.3
70.0
21.9
65.6
23.8
71.4
24.6
73.6
25.5
76.4
7.1
23.7
71.0
22.2
66.6
24.1
72.4
24.9
74.7
25.8
775
7.2
24.0
72.0
22.5
67.5
24,5
73.4
25.3
757
26.2
78.6
7.3
24.3
730
22.8
68.4
24.8
74.5
25.6
76.8
26.6
IH
7.4
24.7
74.0
23.1
69.4
25.2
75.5
26.0
77.9
26.9
80.8
7.5
25.0
75.0
^H
70.3
25.5
76.5
26.3
78.9
27.3
81.9
7.6
253
76.0
23.8
71.3
25.9
775
26.7
80.0
27.7
83.0
7.7
25.7
77.0
24.1
72.2
26.2
78.5
27.0
81.0
28.0
84.1
7.8
26.0
78.0
24.4
73.1
26.5
79.6
27.4
82.1
^?i
85.2
7.9
26.3
79.0
24.7
74.1
26.9
80.6
27.7
83.1
28.8
86.3
8.0
26.7
80.0
25.0
75.0
27.2
81.6
28.1
84.2
29.1
87.4
8.1
27.0
81.0
253
759
27.5
82.6
28.4
85.2
295
88.5
8.2
27.3
82.0
25.6
76.9
27.9
83.6
28.8
86.3
29.9
89.5
8.3
27.7
83.0
25.9
77.8
28.2
84.7
29.1
111
302
90.6
8.4
28.0
84.0
26.3
78.8
28.6
85.7
29.5
29.8
306
91.7
!5
28.3
85.0
26.6
79.7
28.9
86.7
89.4
30.9
92.8
8.6
28.7
86.0
26.9
80.6
29.2
ll^
30.2
90.S
313
939
8.8
29.0
5Z°
27.2
81.6
29.6
IH
30.5
91.5
31.7
95.0
29.3
88.0
27.5
82.5
29.9
89.8
30.9
92.6
32.0
96.1
8.9
29.7
89.0
27.8
83.4
30.3
90.8
31.2
936
32.4
97.2
9.0
30.0
90.0
28.1
84.4
30.6
91.8
316
94.7
32.8
98.3
9.1
30.3
91.0
28.4
!l3
309
92.8
319
957
33.x
99.4
9.2
30.7
92.0
28.8
86.3
313
93.8
32.3
96.8
33.5
100.5
9.3
31.0
930
29.1
87.2
31.6
94.9
32.6
97.8
33.9
101.6
9.4
31.3
94.0
29.4
88. z
32.0
959
33.0
98.9
342
102.7
^•5
31.7
95.0
29.7
89.1
32.3
96.9
333
99.9
346
X03.7
9.6
32.0
96.0
30.0
90.0
32.6
97.9
33.7
lOI.O
34.9
104. S
H
32.3
97.0
30.3
90.9
33'0
98.9
340
102.0
35.3
105.9
9.8
32.7
98.0
30.6
91.9
33.3
100.
34.4
103. 1
35.7
107.0
9.9
33.0
99.0
30.9
92.8
33.7
lOI.O
34.7
104.2
36.0
108. 1
lO.O
333
100.
31.3
93.8
340
102.0
35.1
105.2
36.4
109.2
See note at foot of page 220.
RULES FOR WEIGHT. 223
Rules for the Weight of Iron and Steel.
The following rules for finding the weight of wrought iron, cast iron,
and steel, are based on the data contained in Tables No. 70 and 71.
Rule i. — ^To find the Weight of Iron or Steel, when the volume
in cubic feet is given. Multiply the volume by
4.29 for wrought iron,
4.02 for cast iron,
4.37 for steel.
The product is the w«ight in hundredweights.
Rule 2. — When the volume in cubic inches is giveny multiply the volume
by
.278 (or .28) for wrought iron,
.26 for cast iron,
.283 for steel.
The product is the weight in pounds.
Rule 3. — WAen the quantity is reduced to square feet, one inch in thickness^
multiply the area by
40 for wrought iron,
37/^ for cast iron,
40.8 (or 41) for steel.
The product is the weight in pounds.
Or, multiply the area by
•357 ^or wrought iron,
•335 ^or cast iron,
.364 for steel.
The product is the weight in hundredweights.
Rule 4. — When the sectional area in square inches, and t/ie length in feet^
of a bar or prism are givcfi, multiply the sectional area by the length,
and by
3 V3 for wrought iron,
3^ for cast iron,
3.4 for steel.
The product is the weight in pounds.
For large masses, multiply the sectional area by the length, and divide
the product by
672 for wrought iron,
717 for cast iron,
659 for steel.
The quotient is the weight in tons.
Rule 5. — W/ien the sectional area in square inches , and the length in yards ,
of a barorprisniy are given, multiply the sectional area by the length, and by
10 for wrought iron,
9.375 for cast iron,
10.2 for steel.
The product is the weight in pounds.
224 WEIGHT OF METALS.
Rule 6. — To find the sectional area of a bar or prism of iron
OR steel, when the length and tfu toted weight are given. Divide the weight
in pounds by the length in feet, and by
3 ^3 for wrought iron,
3^ for cast iron,
3.4 for steel.
The quotient is the sectional area in square inches.
Rule 7. — To find the length of a bar, prism, or other piece
of uniform section of iron or steel, whefi the total weight and the
sectional area are given. Divide the weight in pounds by the sectional area
in square inches, and by
3 '/3 for wrought iron,
3 ^ for cast iron,
3.4 for steel.
The quotient is the length in feet.
In applying the last rule to calculate the length of wire of a given size,
for a given weight, say i cwt. of wire, the sectional area of the wire is
found, in the usual way, by multiplying the square of the thickness or
diameter, //, by .7854. Then, by the rule, the length in feet of i cwt. of
iron wire is equal to
112 42.78
In the same way, the dividends of the fractions to express the length of
I cwt. of other metals may be found, and the following is a special rule
lor wire : —
Rule 8. — To find the length of one hundredweight of wire
OF A given thickness. Divide the following numbers by the square of
the diameter or thickness, in parts of an inch : —
42.78 for wrought iron,
42 for steel,
37.43 for copper,
38.54 for brass,
31.34 for silver,
17.12 for gold,
15.28 for platinum.
The quotient is the length in feet.
Note. — This rule may be used for finding the weight of round bar iron.
2. It is known that the density of wire is not perfectly constant, but
that there is some degree of variation, according to the size. It is generally
understood that the density is reduced as the wire is drawn smaller, but
it appears from the table of the weight of Warrington wire, that the density
is greater as the size is less. The same inference is to be drawn from
tabular statements of the length of one kilogramme of wire according to
the French gauge (Table No. 31, page 148). One of these statements is
given on the next page, from which it is apparent that the length of iron
RULES FOR WEIGHT.
225
required to weigh a kilogramme decreases more rapidly than the sectional
area increases. For example, the diameter being
6, 12, 24, 30 tenths of a millimetre,
the squares of which, or the relative volumes of a given length, are as
I. 4, 16, 25;
the lengths of a kilogramme are
405» ii5> 3o> 20 metres,
which are inversely as
^ 35» i35> 20.2.
Showing that a shorter length is required in proportion to the volume, as
the diameter of the wire is reduced, and that the density of the smaller
wire must therefore be the greater.
Table No. 73^. — ^Weight of Galvanized Iron Wire (French).
Ka. of Gauge.
Diameter.
Length of
X Kilogramme. 
No. of Gauge.
Diameter.
Length of
z Kilogramme.
millimetres.
metres.
millimetres.
metres.
I
0.6
405
13
0.20
40
2
0.7
370
14
0.22
35
3
0.8
260
15
0.24
30
4
0.9
215
16
0.27
25
5
O.IO
175
17
0.30
20
•
6
O.I I
140
18
0.34
15
7
0.12
115
19
039
10
8
0.13
103
20
0.44
9
9
0.14
82
21
0.49
6
10
015
70
22
0.54
5
II
0.16
65
23
059
4
12
0.18
50
3. The densities of metals assumed in the foregoing rules are those which
are tabulated in Table No. 65.
4. In estimating the weight of cast iron from plans, the weight is fre
quently calculated at the same rate as for wrought iron, which is heavier
^n cast iron, with the object of providing an allowance, by way of com
pensation, for occasional swellings or enlargements of castings in excess of
'^''•e exact dimensions of patterns.
The following tables of the weight of metals in various forms have been
'Collated by means of the preceding rules. The sectional areas of bars
~id other pieces of uniform section are, in some tables, added for each
^^ntling. The length of bar, and the area of plates and sheets, required
'"^ weigh I cwt., or i ton, are given.
10
226 WEIGHT OF METALS.
LIST OF TABLES OF THE WEIGHT OF WROUGHT IRON,
In Bars, Plates, Sheets, Hoopiron, Wire, and Tubes.
Table No. 74. — Weight of Flat Bar Iron; width, i to 11 inches; thick
ness, 7x6 to I inch; length, i to 9 feet.
Table No. 75. — Weight of Square Iron; ^ to 6 inches square; length,
I to 9 feet
Table No. 76. — ^Weight of Round Iron, ^ to 24 inches in diameter;
length, I to 9 feet
Table No. 77.— Weight of AngleIron and TeeIron; sum of the width
and depth, ly^ to 20 inches; thickness, J^ to i inch; length, i foot
In the composition of this table, it has been assumed that the base and
the web or flange are of equal thicknesses; and that the reduction of area
of section by rounding off the edges, is compensated by the filling in at
the root of the flange.
Table No. 78. — ^Weight of Wroughtiron Plates; area, i to 9 square
feet; thickness, X to 15 inches.
Table No. 79. — Weight of Sheet Iron, according to wiregauge used by
South Staffordshire sheetrollers; area, i to 9 square feet; thickness. No. i
to No. 32 wiregauge.
Table No. 80. — Weight of Black and Galvanized Iron Sheets (Morton's
Table).
Table No. 81. — Weight of Hoop Iron; width, ^ to 3 inches; thickness.
No. 4 to No. 21 wiregauge; length, i foot
Table No. 82. — ^Weight and Strength of Warrington Iron Wire.
Table No. 83. — Weight of Wroughtiron Tubes, by internal diameter;
diameter, f^ to 36 inches; thickness, ^ inch to No. 18 wiregauge; length,
I foot
. Table No. 84. — Weight of Wroughtiron Tubes, by external diameter;
diameter, i to 10 inches; thickness, No. 15 wiregauge to s/x6 inch; length,
I foot
Multipliers, derived from table No. 70, are subjoined, by which the
tabulated weights of wrought iron may be multiplied, in order to find from
these tables the weight of bars, plates, or sheets of other metal. —
Multipliers.
Hammered Iron i.oi
Cast Iron 94
Steel ; 1.02
Sheet Copper 1.14
Hammered Copper 1.16
Lead 1.48
Cast Brass 1.05
Brass Wire i.ii
Gun Metal 1.09
FLAT BAR IRON.
227
Table No. 74— WEIGHT OF FLAT BAR IRON.
I INCH Wide.
Thick
xass.
Sbct.
AXSA.
Length in ]
Feet.
Len^h
to weigh
X cwt.
z
2
3
4
5
6
7
8
9
iacbes.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
ll».
lbs.
lbs.
lbs.
feet
H
5/16
H
llt6
250
.313
.438
.500
.833
1.04
1.25
1.46
1.67
1.67
2.08
2.50
2.92
333
2.50
3.12
375
4.38
5.00
3.33
4.16
5.00
5.84
6.67
4.17
5.20
6.25
7.29
8.33
5.00
6.24
7.50
8.76
lO.O
5.83
7.28
8.75
10.2
II.7
6.67
8.32
10.
II.7
133
7.50
9.36
"3
13. 1
15.0
134.4
89.6
76.8
67.2
9/t6
"/i6
•5^3
.750
1.88
2.08
2.29
2.50
3.75
4.16
4.58
5.00
5.62
6.25
6.87
7.50
7.50
8.33
9.17
lO.O
9.37
10.4
II.4
12.5
"•3
12.5
13.8
15.0
13 1
14.6
16.0
17.5
15.0
16.6
18.3
20.0
16.9
18.8
20.6
22.5
597
538
48.9
44.8
I
.813
.875
938
1. 00
2.71
2.92
3.13
3.33
5.42
5.84
6.25
6.67
8.12
8.76
9.38
10.0
10.8
II. 7
12.5
13.3
13.5
14.6
15.6
16.7
16.3
18.8
20.0
19.0
20.4
21.9
23.3
21.7
23.4
25.0
26.7
24.4
26.3
28.1
30.0
41.4
38.4
35.8
33.6
i}i INCHES Wide.
1
iacbes.
sq. in.
V
.2S1
S/jfi
.352
H
.422
7/16
.492
H
.563
»/t«
.633
H
.703
r
.844
r
.914
.984
'5/16
1.06
I
I.I3
lbs.
.938
I.17
I.4I
1.64
1.88
2. II
2.34
2.58
2.91
3.05
3.28
3.52
375
lbs.
1.88
2.34
2.S1
3.28
3.75
422
4.69
5.16
5.63
6.09
6.56
7.03
7.50
lbs.
2.81
3.52
4.22
4.92
5.62
6.33
7.03
7.73
8.44
9.14
9.84
10.6
"•3
lbs.
3.75
4.68
5.62
6.56
7.50
8.44
9.38
10.3
"3
12.2
13. 1
14. 1
150
lbs.
4.69
5.86
7.03
8.20
9.38
10.6
n.7
12.9
14.0
15.2
16.4
17.6
18.8
lbs.
5.63
7.03
8.44
9.84
"3
12.7
13.1
15.5
16.9
18.3
19.7
21. 1
22.5
lbs.
6.56
8.20
9.84
II.5
13.1
14.8
16.4
18.0
19.7
21.3
23.0
24.6
26.3
lbs.
7.50
9.37
"•3
13. 1
15.0
16.9
18.8
20.6
22.5
24.4
26.3
28.1
30.0
lbs.
8.44
10.6
12.7
14.8
16.9
19.0
21. 1
23.2
25.3
27.4
29.5
31.6
33.8
1% INCHES Wide.
lbs.
3.12
3.91
4.69
5.47
6.25
7.03
7.81
8.59
9.38
10.2
10.9
II. 7
12.5
lbs.
4.17
5.21
6.25
7.29
8.33
9.38
10.4
II.5
12.5
13.5
14.6
15.6
16.7
lbs.
5.21
6.51
7.81
9.12
10.4
II.7
13.0
14.3
15.6
16.9
18.2
20.8
lbs.
6.25
7.82
9.38
10.9
12.5
14. 1
15.6
17.2
18.8
20.3
21.9
23.4
25.0
lbs.
7.29
9. 1 1
10.9
12.8
14.6
16.4
18.2
20.1
21.9
237
255
27.3
29.2
lbs.
8.33
10.4
12.5
14.6
16.7
18.8
20.8
22.9
25.0
27.1
29.2
31.2
33.3
lbs.
9.37
II. 7
14. 1
16.4
18.8
14. 1
23.4
2J.8
28.1
30.5
32.8
35.1
37.5
feel.
II9.5
95.6
79.6
68.3
59.7
531
47.8
434
398
36.8
34.1
31.9
29.9
feet.
107.5
94.0
71.7
61.2
53.8
47.8
43.0
39.1
35.8
33.'
30.7
28.7
26.9
228
WEIGHT OF METALS.
Weight of Flat Bar Iron.
i)i INCHES Wide.
Length in Feet.
Thick
ness.
Sect.
Area.
Len^
to weigh
z
a
3
4
5
6
7
8
9
X CWL
inches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
X
.344
1.15
2.29
3.44
4.58
5.73
6.87
8.02
9.17
10.3
97.7
S/i6
.430
143
2.86
4.30
5.73
7.16
8.59
10.
II.5
12.9
78.2
H
.516
1.72
344
5.16
6.87
8.59
10.3
12.0
137
'$■5
65.6
7/16
.602
2.01
4.01
6.02
8.02
10.
12.0
14.0
16.0
18.0
48."9
}i
.688
2.29
4.58
6.87
9.17
"•5
138
16.0
18.3
2a 6
9/16
•273
.859
2.58
2.86
5.16
7.73
10.3
12.9
15.5
18.0
20.6
1
23.2
434
H
5.73
8.59
II.5
'43
17.2
20.1
22.9
25.8
39.1
r
.945
315
6.31
6.88
945
12.6
15.8
18.9
22.1
25.2
28.4
355
1.03
3.44
10.3
13.8
17.2
20.6
24.1
27.5
309
•32.6
»3/i6
1. 12
372
8.02
II. 2
14.9
18.6
22.3
26.1
29.8
33.5
30.1
'A
1.20
4.01
12.0
16.0
20.0
24.1
28.1
32.1
36.1
27.9
»s/i6
1.29
4.30
8.59
12.9
17.2
21.5
25.8
30.1
344
38.7
26.1
I
i.3»
4.58
9.17
13.8
18.3
22.9
27.5
32.1
36.7
41.3
24.4
i}i INCHES Wide.
inches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
X
.275
1.25
•2.50
375
5.00
6.25
7.50
8.75
10.
"•3
89.6
s/16
.469
1.56
3.13
4.69
6.25
7.82
9.38
10.9
125
14. 1
78.3
H
.563
1.88
375
5.63
7.50
8.75
9.38
IL3
13. 1
150
16.9
59.7
7/16
.656
2.19
4.38
6.56
10.9
13. 1
15.3
17.5
19.7
5*;
H
.750
2.50
5.00
7.50
10.0
12.5
15.0
17.5
20.0
22.5
44.8
9/16
.844
2.81
5.63
8.44
"•3
14. 1
16.0
19.7
22.5
25.3
39.8
^,
.938
3.13
6.25
6.88
9.38
12. J
15.6
18.8
21.9
25.0
28.1
35.8
r
1.03
344
10.3
13.8
17.2
20.6
24.1
27.5
30.9
32.6
i»3
3.75
7.50
"3
15.0
18.8
22.5
26.3
300
33.8
29.9
«3/i6
1.22
4.06
8.13
12.2
16.3
20.3
24.4
28.4
32.5
36.6
27.6
^.
I3I
4.38
8.75
131
17.J
21.9
26.3
30.6
350
39.4
25.6
»s/i6
1.41
4.69
938
14. 1
18.8
23.4
28.1
32.8
37.5
42.2
23.9
I
1.50
5.00
lO.O
15.0
20.0
25.0
30.0
350
40.0
45.0
22.4
i^ INCHES Wide.
inches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
X
.406
135
2.71
4.06
541
6.8
8.10
9.48
10.8'
12.2
82.7
s/16
.508
1.69
339
5.07
6.77
8.5
10.2
II.8
13.5
15.2
66.2
H
.boq
2.03
4.06
6.09
8.12
10.2
12.2
14.2
16.2
183
55.1
7/16
.813
2.37
474
7.II
8.12
9.48
11.8
14.2
16.6
19.0
21.3
47.3
^
2.71
5.42
10.8
135
16.2
19.0
21.6
244
41.3
9/16
.914
3.05
6.09
9.14
12.2
^i^
18.3
21.3
24.4
27.4
36.8
^,
1.02
339
6.77
10.2
13.5
16.9
20.3
23.7
^H
30.5
33.1
"A6
1. 12
372
7.45
II. 2
14.9
18.6
22.3
26.1
29.8
335
30.1
H
1.22
4.06
8.13
12.2
16.3
20.3
24.4
28.4
32.5
36.6
27.6
»3/i6
1.32
4.40
8.80
132
17.6
22.0
26.4
30.8
352
39.6
25.4
^.
L43
4.74
9.48
14.2
19.0
23.7
28.4
33.2
37.9
42.7
23.6
^s/t6
I.S3
5.08
10.2
15.2
20.3
254
30.5
35.5
40.6
48.8
22.1
I
1.63
5.42
10.8
16.3
21.7
27.1
32.5
379
43.3
21.2
FLAT BAR IRON.
229
Weight of Flat Bar Iron.
1^ INCHES Wide.
Length in
Fbbt.
Thioc
NESS.
Sbct.
Aksa.
Length
to weigh
Z
2
3
4
5
6
7
8
9
X cwt.
inches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet
X
.638
1.46
2.92
4.37
5.83
7.29
8.74
10.2
II.7
13. 1
76.8
5/16
.547
1.82
3.6J
§^?
7.29
9.II
10.9
12.8
14.6
16.4
61.4
H
.656
2.19
4.38
6.56
8.75
10.9
13.1
153
ns
19.7
51.2
7/16
.766
2.55
H^
7.66
10.2
12.8
15.3
17.9
20.4
23.0
43.9
a
.875
2.92
5.«3
8.75
II.7
14.6
17.5
20.4
23.3
26.2
38.4
9/16
.984
3.28
6.56
9.84
13. 1
16.4
19.7
23.0
26.2
29.5
34.1
H
1.09
3.65
7.29
10.9
14.6
19.2
21.9
2$5
29.2
32.8
•307
r
1.20
4.01
8.02
12.0
16.0
20.0
24.1
28.1
32.1
36.1
27.9
I.3I
4.3«
8.75
13.1
17.5
21.9
26.3
306
350
39.4
25.6
r
1.42
1.53
4.74
5.10
9.48
10.2
14.2
15.3
19.0
20.4
23.7
25. S
28.4
30.6
33.2
357
37.9
40.8
432
45.9
23.7
21.9
^s/i6
1.64
547
10.9
16.4
21.9
27.3
32.8
1? ) 1
40.8
437
49.2
20.5
I
1.75
5.«3
II. 7
17.5
233
29.2
35.0
46.7
52.5
19.2
iji INCHES Wide.
tnf<^^
1
sq. in.
' .469
lbs.
1.56
lbs.
3.13
lbs.
4.69
lbs.
6.2$
lbs.
7.81
lbs.
9.38
lbs.
10.9
lbs.
12.5
lbs.
14. 1
feet
71.7
'4
5/.6
1 .586
1.95
391
5.86
7.81
9.66
11.7
13.7
15.6
17.6
573
> H
.703
2.34
4.69
7.03
9.37
II.7
14. 1
16.4
18.8
21. 1
47.8
r/i6
1 .820
2.73
5.47
8.20
10.9
13.7
16.4
19.1
21.9
24.6
41.0
}i
1 .938
313
6.25
9.38
12.5
156
18.8
21.9
25.0
28.1
35.8
9/16
1.06
3.52
7.03
10.5
14. 1
17.6
21. 1
24.6
28.1
31.6
31.8
H
1. 17
39"
7.81
II.7
14.6
195
23.4
27.3
31.2
35.2
28.7
"/•«
1.29
430
8.59
12.9
17.2
21.5
2J.8
28.1
30.1
34.4
38.7
26.1
' )i
I.4I
4.69
9.38
14. 1
18.8
234
32.8
37.5
42.2
239
: »Vi6
! 1.52
5.08
10.2
15.2
20.3
25.4
30.5
355
40.6
45.7
22.1
H
1 1.64
t^
10.9
16.4
21.9
27.3
32.8
38.3
43.9
49.4
20.5
1 '5/i«
1.76
11.7
17.6
23.4
29.3
35.1
41.0
46.9
52.7
19. 1
1'
J.88
6.25
12.5
18.8
25.0
3>.3
37.5
43.8
50.0
56.2
17.9
2 INCHES Wide.
indies.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet
j(
.500
1.67
333
5.00
6.67
8.33
10.0
II.7
13.3
15O
67.2
s/i6
.625
2.08
4.17
6.25
8.33
10.4
12.5
14.6
16.7
18.8
53.8
H
1 .750
2.50
5.00
7.50
lO.O
12.5
15.0
17.5
20.0
22.5
44.8
7/.6
875
2.92
5.83
8.75
II.7
14.6
17.5
20.4
23.3
26.3
38.4
>^
I.OO
333
6.67
10.0
133
16.7
20.0
23.3
26.7
300
336
9/i«
1 1. 13
3.75
7.50
"3
15.0
18.8
22.5
26.3
300
338
29.9
X,
1 1.38
4.17
8.33
12.5
13.8
16.7
20.8
25.0
29.2
33.3
37.5
26.9
r
4.58
9.16
18.3
22.9
27.5
32.1
36.7
41.2
24.4
1.50
5.00
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
22.4
«3/«6
' 1.63
5J^
10.8
16.3
21.7
27.2
32. s
37.9
43.3
48.8
20.7
H
m
S.83
11.7
'75
18.8
23.3
29.2
35.0
40.8
46.7
52.5
19.2
«5/x6
6.25
12.5
25.0
31.3
37.5
43.8
50.0
56.3
17.9
I
2.00
6.67
13.3
20.0
26.7
33.3
40.0
46.7
52.2
60.0
16.8
230
WEIGHT OF METALS.
Weight of Flat Bar Iron.
2>i INCHES Wide.
Length in
Febt.
Thick
ness.
Sect.
Area.
Len^h
to weigh
z
2
3
4
5
6
7
8
9
I cwt.
inches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet
V
.531
1.77
3.54
I'V
7.08
8.85
10.6
12.4
14.2
15.9
63.2
5/i«
.664
2.21
4.43
6.64
8.85
II.7
13.3
^55
17.7
19.9
50.6
H
.797
2.66
531
7.97
10. D
13.3
'59
i8.6
21.2
23.9
42.2
7/16
.930
3.10
6.20
930
12.4
15.5
18.6
21.7
24.8
27.9
36.1
}i
1.06
3.54
7.08
10.6
14.2
17.7
21.3
24.8
28.3
31.9
31.6
9/16
1.20
3.98
7.97
12.0
15.9
20.0
23.9
27.9
31.9
355
28.1
^.
1.33
4.43
8.85
13.3
17.7
22.1
1 26.6
31.0
35.4
39.8
25.3
r
1.46
4.87
9.74
14.6
19.5
24.4
1 29.2
34.1
39
43.8
23.0
1.59
5.3«
10.6
"59
21.2
26.6
31.9
37.2
42.5
47.8
21. 1
»3/x6
1.74
5.76
11.5
'H
23.0
28.8
34.5
40.3
46.0
51.8
19.8
H
1.86
6.20
12.4
18.6
24.8
31.0
37;
43.4
49.6
55.8
18. 1
»5/i6
1.98
6.64
13.3
19.9
26.6
33.2
39.8
46.5
53.1
59.8
16.9
I
2.13
7.08
14.2
21.3
28.3
35.4
42.5
49.6
56.7
63.8
15.8
2)4 INCHES Wide.
2f^ inches Wide.
inches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
X
.563
1.88
3.75
5.63
7.50
9.4
"3
131
15.0
16.9
59.7
5/16
.703
2.34
4.69
7.03
9.38
II.7
14. 1
16.4
18.8
21.1
47.8
^
.844
2.81
5.63
8.44
"3
14. 1
16.9
19.7
22.5
253
39.8
7/16
.984
3.28
6.56
9.84
13. 1
16.4
19.7
23.0
26.3
29.5
33.8
34.1
^
I.I3
3.75
7.50
11.3
15.0
18.8
22.5
26.3
30.0
29.9
9/16
1.27
4.22
8.44
12.7
16.9
21. 1
^53
29.5
33.8
38.0
26.5
^,
I.4I
4.69
9.38
14. 1
18.8
23.4
28.1
32.8
37.5
42.2
23.9
r
1.55
5.16
10.3
15.5
20.6
25.8
30.9
36.1
41.3
46.4
21.7
1.69
5.63
"3
16.9
22.5
28.1
33.8
39.4
4S.O
50.6
19,9
»3/i6
1.83
6.09
12.2
18.3
24.4
30.5
36.6
42.7
48.8
54.9
18.4
^.
1.97
6.56
13. 1
19.7
26.3
32.8
39.4
45.9
52.5
59.1
17.I
»5/i6
2.II
7.03
14.1
21. 1
28.1
35.2
42.2
49.2
56.3
63.3
15.9
I
2.25
7.50
15.0
22.5
30.0
37.5
45.0
52.5
60.0
67.5
14.9
inches.
5/16
H
7/16
9/x6
"As
H
'3/16
H
«s/x6
sq. in.
.594
.742
.891
1.04
1. 19
134
1.48
1.67
1.78
.08
19
2.2
2.3
lbs.
1.98
2.47
2.97
3.46
396
4.45
4.95
544
594
6.43
6.93
7.42
7.92
lbs.
3.96
4.95
5.94
6.93
7.92
8.91
9.90
10.9
II. 9
12.9
13.9
14.8
15.8
lbs.
594
7.42
8.91
10.4
II. 9
134
14.8
16.3
17.8
19.5
20.8
22.3
23.8
lbs.
7.92
9.90
II.9
139
15.8
17.8
19.8
21.8
23.8
257
27.7
29.7
31.7
lbs.
9.90
12.4
14.8
'73
19
i
22.3
24.7
27.2
29.7
32.2
346
37.1
39.6
lbs.
II. 9
14.8
17.8
20.8
23.8
26.7
29.7
32.7
35.6
38.6
41.6
44.5
47.5
lbs.
139
17.3
20.8
24.2
277
31.2
346
38.1
41.6
45 o
48.5
5".9
554
lbs.
15.8
19.8
23.8
27.7
3J.7
356
39.6
435
47.5
51.5
554
59.4
63.3
lbs.
17.8
22.3
26.7
312
356
40.1
43.5
49.0
534
57.9
62.3
66.8
71.3
feet.
56.6
45.3
37.7
32.3
28.3
25.2
22.6
20.6
18.9
17.4
16.2
15.1
14.2
FLAT BAR IRON.
231
Weight of Flat Bar Iron.
2% INCHES Wide.
'
Length in ]
Fekt.
T *t
Thick
Sbct.
Area.
Len^h
to weigh
KESS.
X
a
3
4
5
6
7
8
9
I cwt.
iaches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
%
.625
2,08
4.17
6.25
8.33
10.4
12.5
14.6
16.7
18.8
53.8
5/16
.781
2.60
5.21
7.81
10.4
13.0
15.6
18.2
20.8
23.4
28.1
43.0
H
.938
3.>3
6.25
9.38
'25
15.6
18.8
21.9
25.0
35.8
7/16
1.09
3.65
7.29
10.9
14.6
18.2
21.9
25.5
29.2
32.8
30.7
'A
1.25
4.17
8.33
12.5
16.7
20.8
25.0
29.2
33.3
37.5
26.9
9/t6
I.4I
4.69
9.38
14. 1
18.8
23.4
28.1
32.8
375
42.2
23.9
H
1.56
5.21
10.4
15.6
20.8
26.0
3».3
36.5
417
46.9
21.5
"A6
1.72
S73
II.5
17.2
22.9
28.6
34.4
40.1
45.8
51.6
19.6
H
I.S8
6.25
12.5
18.6
25.0
31.3
37.5
43.8
50.0
56.3
18.0
n/i6
2.03
6.77
13.5
20.3
27.1
33.8
40.6
47.4
54.2
60.9
16.5
'^,
2.19
7.29
14.6
21.9
29.2
36.5
43.7
51.0
58.3
65.7
15.4
iS/i6
234
7.81
15.6
23.4,
31.3
39.0
46.9
54.7
62.5
70.3
143
I
1
2.50
8.33
16.7
25.0
333
41.7
50.0
58.3
66.7
75.0
13.4
2^ INCHES Wide.
tnrbrf.
sq. in.
.656
lbs.
2.19
lbs.
4.38
lbs.
6.56
lbs.
8.75
lbs.
10.9
lbs.
13.1
lbs.
15.3
lbs.
17.5
lbs.
19.7
feet.
5>.2
J4,
5/.6
.820
2.73
547
8.20
10.9
13.7
16.4
19. 1
21.9
24.6
41.0
H
.984
3.28
6.56
9.84
13.1
16.4
19.7
23.0
26.2
29.5
34.2
7/16
I.I5
3. 8 J
7.66
II.5
15.3
19. 1
23.0
26.8
30.6
34.4
29.3
}i
I.3I
4.38
8.75
13.1
17.5
21.9
26.3
30.6
3S0
39.4
25.6
,/;6
1.48
4.92
9.84
14.8
19.7
24.6
29.5
345
394
44.3
22.8
^,
1.64
547
10.9
16.4
21.9
27.3
32.8
38.3
43.8
49.2
20.5
"A«
I.8I
6.02
12.0
18. 1
24.1
30.2
1 36.1
42.1
48.1
54.1
18.6
H
1.97
6.56
13. 1
19.7
26.3
32.8
1 39.4
45.9
52.5
59.1
17. 1
•3/16
2.13
7.11
14.2
21.3
28.4
355
1 42.7
49.8
56.9
64.0
15.8
^.
2.30
7.66
153
23.0
30.6
38.3
! 45.9
536
61.3
68.9.
14.7
»s/lfi
2.46
8.20
16.4
24.6
32.8
41.0
I 49.2
57.4
65.6
73.8
13.7
I
2.63
8.75
17.5
26.3
35.0
43.8
52.5
61,3
70.0
78.8
12.8
2^ INCHES Wide.
1
indies.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
X
.688
2.29
4.58
6.87
9.17
II.5
13.8
15.0
18.3
20.6
48.9
5/x6
.859
2.86
573
8.59
II. 5
14.3
17.2
20.1
22.9
25.8
39.1
H
1.03
344
6.88
10.3
13.8
17.2
20.6
24.1
27.5
309
32.8
7/16
1.20
4.01
8.02
12.0
16.0
20.1
24.1
28.1
32.1
36.1
27.9
}^
138
4.58
9.17
13.8
18.3
22.9
27.5
32.1
36.7
41.3
24.4
9/16
1.55
5.16
IO3
15.5
20.6
25.8
30.9
36.1
41.3
46.4
21.7
H
1.72
5.73
11.5
17.2
22.9
28.6
34.4
40.1
45.8
51.6
*95
"/t6
1.89
6.30
12.6
18.9
25.2
31.5
37.8
44.1
50.4
56.7
17.8
M
2.06
6.88
138
20.6
27.5
34.4
41.3
48.1
55.0
61.9
16.3
n/16
2.23
7.45
14.9
22.3
29.8
37.2
44.7
52.1
59.6
67.0
15.0
Ji
2.41
8.02
16.0
24.1
32.1
40.1
48.1
56.1
64.2
72.2
14.0
*S/x6
2.58
8.59
17.2
25.8
344
430
51.6
60.1
68.8
77.3
13.0
1
2.75
9.17
18.3
27.5
36.7
45.8
55.0
64.2
73.3
82.5
12.2
232
WEIGHT OF METALS.
Weight of Flat Bar Iron.
2% INCHES Wide.
Length in ]
Feet.
Thick
ness.
Sect.
Area.
Length
to weigh
z
2
3
4
5
6
7
8
9
X cwt.
lAches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
%,
.719
2.40
4.79
7.19
9.58
12.0
14.4
16.8
19.2
21.6
46.7
5/16
.898
3.00
6.cx>
9.00
12.0
15.0
18.0
21.0
24.0
27.0
37.4
H
1.08
3.59
7.19
10.8
14.4
iS.o
21.6
25.2
28.8
32.3
312
7/16
1.26
4.19
8.39
12.6
16.8
21.0
25.2
29.4
335
37.7
26.7
'A
1.44
479
9.58
14.4
19.2
24.0
28.8
335
38.3
43.1
23.4
9/16
1.62
539
10.8
16.2
21.6
27.0
32.3
37.7
^3'
48.5
20.8
^,
1.80
599
12.0
18.0
24.0
300
36.0
42.0
48.0
54,0
18.7
"A6
1.98
6.59
13.2
19.8
26.4
33.0
40.5
46.1
52.7
59.3
17.0
H
2.16
7.19
14.4
21.6
28.8
36.0
43.1
50.3
57.5
64.7
1
15.6
t'
2.34
7.79
^H
23.4
311
39.0
46.7
54.5
62.3
70.1
14.4
2.52
8.39
16.8
25.2
33.5
42.0
503
58.7
67.1
755
13.4
»5/i6
2.70
8.98
18.0
27.0
35.9
45.0
53.9
62.9
71.9
80.9
12.4
I
2.88
9.58
19.2
28.8
38.3
48.0
575
67.1
76.7
86.3
II.7
3 INCHES Wide.
inches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
1
lbs.
lbs.
lbs.
lbs.
feet.
%
.750
2.50
5.00
7.50
lO.O
12.5
15.0
17.5
20.0
22.5
44.8
s/i6
.938
313
6.25
9.38
I2.S
'^Z
18.8
21.9
25.0
28.1
35.8
H
1. 13
3.75
7.50
II3
15.0
18.8
22.5
26.3
30.0
33.8
29.9
7/i6
I3I
438
8.75
131
17.5
21.9
26.3
30.6
350
39.4
25.6
'A
1.50
5.00
lO.O
15.0
20.0
25.0
30.0
35.0
40.0
45.0
22.4
9/16
1.69
563
"3
16.9
22.5
28.2
33.8
39.4
450
50.6
.19.9
H
1.88
6.25
12.5
18.8
25.0
313
37.5
43.8
50,0
56.3
17.9
r
2.06
6.88
13.8
20.6
27.5
34.4
41.3
48.1
55.0
61.9
16.3
2.25
7.50
15.0
22.5
30.0
375
45.0
52.5
60.0
67.5
14.9
T
2.44
8.13
16.3
24.4
32.5
40.7
48.8
56.9
65.0
73'
13.8
2.63
8.75
'75
26.3
35.0
43.8
52.5
61.3
70.0
78.8
12.8
^5/16 1
2.81
9.38
18.8
28.1
375
46.9
56.3
65.6
75.0
84.4
12.0
I 1
1
30O
10.0
20.0
300
40.0
50.0
60.0
70.0
80.0
90.0
II. 2
3X inches Wide.
inches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
fecL
X
.813
2.71
5.42
8.13
10.8
13.6
16.3
19.0
21.7
24.4
41.3
5/16
1.02
3.39
6.77
10.2
13.5
16.9
20.3
23.7
27.1
30.5
33.1
H
1.22
4.06
8.13
12.2
16.3
20.3
24.4
28.4
32.5
36.6
27.5
7/16
1.42
4.74
9.48
14.2
19.0
237
28.4
33.2
37.9
4?/Z
23.6
yi
1.63
5.42
10.8
16.3
21.7
27.1
32.5
37.9
433
48.8
20.7
9/.«
1.83
6.09
12.2
18.3
24.4
305
36.6
42.7
48.7
54.8
18.4
H
2.03
6.77
13.5
20.3
27.1
33.9
40.6
47.4
54.2
60.9
16.5
r
2.23
745
14.9
22.3
29.8
37.2
44.7
52.1
59.6
67.0
15.0
2.44
8.13
16.3
24.4
32.5
40.6
48.8
56.9
65.0
731
13.7
r
2.64
8.80
17.6
26.4
35.2
44.0
52.8
61.6
70.4
79.2
12.7
2.84
9.48
19.0
28.4
37.9
47.4
56.9
66.4
75.8
85.3
II.8
»5/i6
3.05
10.2
20.3
30. 5
40.6
50.8
60.9
71. 1
81.2
91.4
II.O
I
325
10.8
21.7
32. 5
43.3
54.2
65.0
75.8
86.7
97.5
10.3
FLAT BAR IRON.
233
Weight of Flat Bar Iron.
S}4 INCHES Wide.
1
Lbngth in
Fbbt.
Thick 1
Sect.
Area.
Lensth
to weigh
I
a
3
4
5
6
7
8
9
I CWL
1 inches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
><■
.875
2.92
5.83
8.75
II.7
14.6
17.5
2G.4
233
26. J
38.4
s/16
1.09
3^5
7.29
10.9
14.6
18.2
21.9
25.5
29.2
32.8
30.7
H
I3I
4.38
8.75
13. 1
17.5
21.9
26.3
30.6
35.0
39.4
25.6
7/16
1.53
5.10
10.2
15.3
20.4
25.5
30.6
35Z
40.8
459
21.9
}i
1.75
5.83
II. 7
17.5
22.3
29.2
35.0
40.8
46.7
52.5
19.2
S/.6 '
1.97
6.56
13. 1
19.7
26.3
32.8
39.4
459
555
5^1
17. 1
H
2.19
7.29
14.6
21.9
29.2
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T
2.41
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3.06
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t
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1.64
1.88
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2.81
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328
3.52
375
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4.69
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8.59
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105.5
25^
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112.5
8.96
4 INCHES Wide.
inchrt.
iq. in.
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feet.
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3.33
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234
WEIGHT OF METALS.
Weight of Flat Bar Iron.
4X INCHES Wide.
Length in
Feet.
Thick
Sect.
Area.
Len^h
to weigh
ness.
I
2
3
4
5
6
7
8
9
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inches.
1
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lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet
V
1.06
3^54
7.08
10.6
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21.3
24.8
28.3
31.9
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4.43
8.85
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26.6
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H
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21.3
26.6
31.9
37.2
42.5
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18.6
24.8
31.0
37.2
434
49.6
55.8
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^
2.13
7.08
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21.3
28.3
35.4
42.5
49.6
56.7
63.8
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ni
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23.9
31.9
39.8
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2.92
9.74
19.5
29.2
39.0
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58.4
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319
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31.9
42.5
531
63.8
744
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3.45
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57.6
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80.6
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103.6
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3.72
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106.2
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4.25
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28.3
42.5
56.7
70.8
85.0
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127.5 1 7.9
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lbs.
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feet
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I.I3
3.75
75,
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18.8
22.5
26.3
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1.69
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1.48
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1.78
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4.75
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317
47.5
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FLAT BAR IRON.
235
Weight of Flat Bar Iron.
5 INCHES Wide.
Length in ]
Feet.
Thick
ness.
Sect.
ASEA.
Len^h
to weigh
z
2
3
4
5
6
7
8
9
I cwt.
ipches.
sq. in.
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lbs.
lbs.
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lbs.
lbs.
lbs.
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feet.
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1.25
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29.2
33.3
37.5
26.9
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5.21
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6.25
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2.81
9.38
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344
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977
3.75
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87.5
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112.5
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4.06
13.5
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40.6
54.2
67.7
81.3
94.8
108.3
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8.27
4.38
14.6
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140.6
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16.7
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116. 7
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sq. in.
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lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet
V
I3I
4.38
8.75
131
17.5
21.9
26.3
32.8
30.6
35.0
394
25.6
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1.64
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10.9
16.4
21.9
27.3
38.3
43.8
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1.97
6.56
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2.63
8.75
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350
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3.28
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indies.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
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feet.
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1.38
4.58
9.17
13.8
18.3
22.9
27.5
32.1
36.7
41.3
24.5
S/t6
1.72
m
11. g
17.2
22.9
28.6
34.4
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45.8
51.6
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2.06
13.8
20.6
27.5
34.4
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48.1
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61.9
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2.41
8.02
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24.1
32.1
40.1
48.1
56.1
64.2
72.2
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2.75
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18.3
27.5
36.7
45.8
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20.6
30.9
41.3
51.6
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3.44
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34.4
45.8
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68.8
80.2
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103. 1
9.77
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3.78
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120.3
137.5
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6.52
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550
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36.7
55.0
733
91.6
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1
128.4
146.7
165.0
6.11
236
WEIGHT OF METALS.
Weight of Flat Bar Iron.
$}( INCHES Wide.
Lbngth in Fbbt.
T *.!_
Thick
ness.
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Length
to weigh
z
2
3
4
5
6
7 .
8
9
X cwt.
inches.
1
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
m:
1.44
4.79
9.58
14.4
19.2
24.0
28.8
33.5
38.3
431
23.4
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1.80
599
12.0
i8.o
24.0
30.0
35.9
41.9
47.9
53.9
18.7
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2.16
7.19
14.4
21.6
28.8
35.9
43.1
50.3
57.5
64.7
15.6
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2.52
8.39
16.8
25.2
33.5
38.3
41.9
50.3
58.7
67.1
755
134
H
2.88
9.58
19.2
28.8
47.9
57.5
67.1
76.7
86.3
II.7
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3.23
10.8
21.6
32.3
43.'
48.0
53.9
64.7
75.5
86.2
97.0
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3.59
12.0
24.0
36.0
60.0
71.9
83.9
95.8
107.8
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r
3.95
13.2
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39.5
52.7
65.9
79.1
92.2
105.4
1 18.6
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28.8
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71.9
86.3
100.6
115.0
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4.67
15.6
31.2
46.7
62.3
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934
109.0
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140.2
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1 17.4
134.2
150.9
6.68
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5.39
18.0
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71.9
89.8
107.8
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161. 7
6.22
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5.75
19.2
38.3
57.5
76.7
95.8
115.0
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153.3
172.5
5.83
6 INCHES Wide.
inches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
H
1.50
5.00
lO.O
15.0
20.0
25.0
30.0
35.0
40.0
45.0
22.4
s/x6
1.88
6.25
12.5
18.8
25.0
31.8
37.5
43.8
50.0
56.3
17.9
H
2.25
7.50
15.0
22.5
30.0
375
45.0
52.5
60.0
67.5
14.9
7/16
2.63
8.75
17.5
26.3
350
43.8
52.5
61.3
70.0
80.0
78.8
12.8
H
3.00
10.0
20.0
30.0
40.0
50.0
60.0
70.0
90.0
II. 2
9/x6
3.38
"3
22.5
33.8
45.0
56.3
67.5
78.8
90.0
101.3
lO.O
H
375
12.5
25.0
37.5
50.0
62.5
75.0
87.5
100.0
112.5
8.96
r
4.>3
4.50
13.8
15.0
27.5
30.0
413
450
60.0
68.8
75.0
82.5
90.0
96.3
105.0
HO.O
120.0
123.7
135.0
8.15
7.47
t
4.88
16.3
32.5
48.8
65.0
81.3
97.5
"37
1300
146.3
6.90
5.25
'75
18.8
35.0
52.5
70.0
87.$
105.0
122.5
140.0
157.5
6.40
>5/i6
5.63
37.5
56.3
75.0
93.8
II2.5
131.3
150.0
168.7
597
I
6.00
20.0
40.0
60.0
80.0
lOO.O
120.0
140.0
160.0
180.0
S.60
6% INCHES Wide.
inches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
%
1.63
542
10.8
16.3
21.7
27.2
32.5
37.9
43.3
49.0
20.7
5/x6
2.03
6.77
13.5
20.3
27.1
339
40.6
47.4
54.2
60.9
16.5
H
2.44
8.13
16.3
24.4
32.5
40.6
48.8
56.9
65.0
73.1
7/x6
2. 84
9.47
18.9
2S.4
37.9
47.4
56.8
66.3
75.8
85.2
14.8
}i
3.25
10.8
21.7
32.5
43.3
54.2
65.0
75.8
86.7
97.5
10.3
9/x6
3.66
12.2
24.4
36.6
48.8
60.9 1
73 '
85.3
975
109.7
9.20
>^.
4.06
13.5
27.1
40.6
54.2
67.7
81.3
94.8
108.3
121.9
8.27
"/i6
447
14.9
29.8
44.7
59.6
74.5
89.4
104.3
1 13.8
119.2
134.1
6.89
H
4.98
16.3
32.5
48.8
65.0
81.3
97.5
130.0
146.3
'3/x6
5.28
17.6
352
52.8
70.4
88.0
105.6
123.2
140.8
158.4
6.36
H
5.68
19.0
379
56.9
75.8
81.3
94.8
1 13.8
132.7
151.7
170.6
591
'5/x6
6.09
20.3
40.6
60.9
101.6
121.9
142.8
162.5
182.8
5.51
I
6.50
21.7
43.3
65.0
86.7
108.3
130.0
1
I5I.7
1733
195.0
5.29
FLAT BAR IRON.
237
Weight of Flat Bar Iron.
7
INXHES Wide.
Length in Feet.
Thick
ness.
Sect.
Area.
Length
to weigh
X
2
3
4
5
6
7
8
9
I CWL
*
inches.
sq. m.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
X
1.75
583
II.7
17.S
23.3
29.2
35.0
40.8
46.7
52.5
19.2
s/i6
2.19
7.29
14.6
21.9
29.2
3^5
43.8
51.0
58.3
65.6
15.4
H
2.63
8.75
17.5
26.3
352
438
52.5
61.3
70.0
78.8
12.8
I ?/««
3o6
10.2
20.4
30.6
40.8
5?/°
61.3
7^5
81.7
91.9
II.O
1 H
3.50
II.7
23.3
350
'46.7
58.3
70.0
81.7
933
105.0
9.60
9/16
394
13. 1
26.3
39.4
5;^ 5
65.6
78.8
91.9
105.0
I18.I
8.53
H
4.38
14.6
29.2
43.8
58.3
72.9
87.5
102. 1
1 16. 7
131.3
7.68
r
481
5.25
16.0
17.5
32.1
35.0
48. 1
52.5
64.2
70.0
80.2
87.5
96.3
105.0
112.3
122.5
128.3
140.0
1444
157.5
6.98
6.40
«3/i6
5.69
19.0
379
56.9
75.8
95.0
II3.8
132.7
151.7
170.6
591
H
6.13
20.4
40.8
61.3
81.7
102. 1
122.5
142.9
163.3
183.8
549
«5/i6
6.56
21.9
43.8
65.6
87.5
109.4
1313
'53 »
175.0
196.9
5.12
I
7.00
23.3
46.7
70.0
93.3
1 16. 7
140.0
163.3
186.7
210.0
4.80
y}^ INCHES Wide.
baches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
X
1.88
6.25
12.5
18.8
25.0
31.3
375
43.8
50.0
56.3
17.9
s/16
2.34
7.81
15.6
23.4
31.3
39. >
46.9
54.7
62.5
70.3
143
H
i 2.81
9.38
18.8
28.1
375
46.9
56.3
65.6
75.0
84.4
II.9
7/16
328
10.9
21.9
32.8
43.8
54.7
65.6
76.6
87.5
98.4
10.2
>i
3.75
12.5
25.0
37.5
50.0
62.5
75.0
87.5
100.
H2.5
8.96
9/16
4.22
14. 1
28.1
42.2
56.3
70.3
84.4
98.4
II2.5
126.6
7.96
H
4.69
15.6
3Jf.3
46.9
62.5
78.1
93.8
109.4
125.0
140.6
7.17
"/16
5.16
"^^'l
344
51.6
68.8
85.9
103. 1
120.3
137.5
154.7
6.52
H
5.63
18.8
37.5
56.3
75.0
93.8
112.5
1313
150.0
168.8
5.97
T
6.09
20.3
40.6
60.9
81.3
I0I.6
121. 9
142.2
162.5
182.8
551
6.56
21.9
43.8
65.6
87.5
109.4
131.3
153.1
175.0
196.9
5.12
»5/i6
703
23.4
46.9
70.3
93.8
117.2
140.6
164. 1
187.5
210.9
4.78
I
7.50
25.0
50.0
75.0
100.
125.0
150.0
175.0
200.0
225.0
4.48
8 INCHES Wide.
inches.
I sq. in.
lbs.
lbs.
lbs.
lbs
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
X
1
2.00
6.67
13.3
20.0
26.7
333
40.0
46.7
533
60.0
16.8
s/16
2.50
8.33
16.7
25.0
33.3
41.7
50.0
58.3
66.7
75.0
13.4
H
' 300
10.
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
II. 2
7/16
350
II.7
233
35.0
46.7
58.3
70.0
81.7
93.3
105.0
0.60
1 ^
4.00
13.3
26.7
40.0
533
66.7
80.0
933
106.7
120.0
8.40
9/16
4.50
15.0
30.0
45.0
60.0
75.0
90.0
105.0
120.0
135.0
7.47
^,
5.00
16.7
33.3
co.o
66.7
83.3
lOO.O
116.7
133.3
150.0
6.72
t'
5.50
18.3
36.7
55.0
73.3
91.7
IIO.O
128.3
146.7
165.0
6.11
6.00
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
180.0
5.60
»3/i6
6.50
21.7
43.3
65.0
86.7
108.3
130.0
151. 7
173.3
195.0
5i7
»,
7.00
233
46.7
70.0
933
116.7
140.0
163.3
186.7
2IOkO
4.80
^5fi6
7.50
25.0
50.0
75.0
100.0
125.0
150.0
175.0
200.0
225.0
4.48
1
8.00
26.7
53.3
80.0
106.7
133.3
160.0
186.7
213.3
240.0
4.20
238
WEIGHT OF METALS.
Weight of Flat Bar Iron.
9 INCHES Wide.
Thicic
NESS.
Sect.
Area.
Length in Fbbt.
Length
to weigh
I cwt.
z
1
3
4
5
6
7
8
9
inches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
5/16
H
7/16
2.25
2.81
3.38
394
4,50
750
9.38
"3
13. 1
15.0
15.0
18.8
22.5
26.3
30.0
22.5
28.1
338
394
450
30.0
375
45.0
60.0
37.5
46.9
tl
65.6
750
45.0
56.3
Hi
78.8
• 90.0
52.5
65.6
78.8
91.9
105.0
60.0
75.0
90.0
105.0
120.0
67.5
84.4
IOI.3
118. 1
135.0
14.9
II.9
10.0
8.53
7.47
9/16
5.06
5.63
6.19
6.75
16.9
18.8
20.6
22.5
33.8
375
413
45.0
50.6
56.3
61.9
67.5
67.5
75.0
82.5
90.0
84.4
938
103. 1
112.5
101.3
112.5
123.8
1350
118. 1
I3I.3
144.4
157.5
1350
150.0
165.0
180.0
151.9
168.8
185.6
202.5
6.64
597
543
4.98
»5/i6
I
8.44
9.00
24.4
26.3
28.1
30.0
48.8
56.3
60.0
73.1
78.8
84.4
90.0
975
105.0
112.5
120.0
121. 9
i3>3
140.6
150.0
146.3
104.5
168.8
180.0
170.6
183.8
196.9
210.0
195.0
210.0
225.0
240.0
219,4
236.3
2531
270.0
4.59
4.26
3.98
3.73
10 INCHES Wide.
inches. '
1
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
^
2.50
8.33
16.7
25.0
333
41.7
50.0
58.3
66.7
75.0
134
S/.6
3.13
10.4
20.8
31.3
41.7
52.1
62.5
72.9
83.3
93.8
10.7
^
3.75
12.5
25.0
375
50.0
62.5
75.0
875
100.0
II2.5
8.96
7/16
438
14.6
29.2
438
58.3
72.9
875
102. 1
116.7
I3I3
7.68
}i
5.00
16.7
333
50.0
66.7
83.3
lOO.O
116.7
133.3
150.0
6.72
9/16
5.63
18.8
37.5
56.3
75.0
93.8
II2.5
13I3
150.0
168.8
597
H
6.25
20.8
^^■7
62.5
833
104.2
125.0
145.8
166.7
187.5
4^89
r
6.88
22.9
45.8
68.8
917
1 14.6
1375
160.4
1833
206.3
7.50
25.0
50.0
75.0
100.
125.0
150.0
175.0
200.0
225.0
4.48
»3/i6
8.13
27.1
54.2
81.3
108.3
1354
162.5
189.6
216.7
243.8
4.14
'^.
8.75
29.2
58.3
87.5
938
1 16. 7
145.8
175.0
^^'i
2333
262.5
3.84
^s/i6
9.40
313
62.5
125.0
156.3
187.5
218.8
250.0
281.3
3.58
I
10.0
333
66.7
100.
133.3
166.7
200.0
2333
266.7
300.0
3.36
II INCHES Wide.
inches.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
^
275
9.17
18.3
27.5
36.7
45.8
§52
s/16
344
11. J
138
22.9
34.4
45.8
68.' i
68.8
H
4. "3
275
413
55.0
82.5
7/16
4.81
16.0
32.1
48.1
64.2
i6.2
96.3
}4
5.50
18.3
36.7
55.0
73.3
91.7
1 10.0
9/16
6.19
20.6
^^•i
61.9
82.5
103,1
123.8
^,
6.88
22.9
458
68.8
91.8
114.6
"375
r
7.56
25.2
50.4
75.6
100.8
126.0
i5>.3
8.25
275
55.0
82.5
IIO.O
137.5
165.0
r
8.94
29.8
596
89.4
II9.2
149.0
178.8
9.63
32.1
64.2
96.3
128.3
160.4
192.5
*5/i6
10.4
344
68.8
103. 1
137.5
171.9
206.3
I
11.
36.7
733
1 10.0
146.7
1833
220.0
lbs. lbs.
642 73.3
80.2 91.7
96.3 I IIO.O
1 12. 3 128.3
128.3 1467
144.4 165.0
160.4 1 183.3
176.5 201.7
192.5 220.0
208.5 238.3
2246 256.7
240.6 I 275.0
256.7 293.3
lbs.
feet.
82.5
12.2
103. 1
123.8
9.77
8.15
144.4
165.0
6.98
6.II
185.6
206.3
226.9
5.43
489
444
2475
4.07
268.1
288.8
3.76
3.49
3094
330.0
3.26
3.06
1
SQUARE IRON.
239
Table No. 75.— WEIGHT OP SQUARE IRON.
Lkngth in Fbbt.
w «
SiDB.
Sbct.
Akba.
Lens;th
to weigh
I
a
3
4
5
6
7
8
9
X cwt.
laches.
aq. b.
lbs.
lbs.
lbs.
lU.
lbs.
lbs.
lbs.
lU.
lbs.
feet.
X
.0156
.052
.104
.156
.208
.260
.313
'l^l
.417
.469
2154
3/16
•035'
.117
.234
•351
.468
.584
.701
.818
.935
1.05
960.0
}(
.0625
.208
.4>7
.625
.833
1.04
1.25
1.46
1.67
1.80
537.6
i/l6
.0977
.326
•^^l
.977
1.30
1.68
'•95
2.28
2.60
2.93
343.8
H
.141
.469
.938
I.4I
1.58
2.34
2.81
3.28
3.75
4.22
238.3
7/,6
! .191
.638
1.28
I.9I
2.55
319
3.83
4.46
5.10
5.74
176.0
H
•25
■833
1.67
2.50
3.33
4.17
5.00
5.83
6.67
7.50
1344
9/l«
.316
1.06
2. II
3 16
4.22
5.27
6.33
7.38
8.44
9.49
106.3
H
•391
1.30
2.60
3.91
5.21
6.51
7.81
9.II
10.4
11.7
85.9
r
, 473
1.58
3.J5
473
6.30
7.88
945
II.
12.6
14.2
71.0
1.563
1.88
375
5.63
7.50
8.80
9.38
"•3
13."
15.0
16.9
59.7
T
.661
2.20
4,40
6.61
II.
132
'54
16.6
19.8
50.8
.766
2.55
5.10
7.66
10.2
12.8
15.3
17.9
20.4
23.0
43.9
*5/i6
,879
2.93
5.86
8.79
II.7
14.7
17.6
20.5
234
26.4
38.2
1
I.CX>
3.33
6.67
10.0
13.3
16.7
20.0
233
26.7
30.0
33.6
I »/i6
. 1.13
3.76
7.53
"•3
15.I
18.8
22.6
26.3
30.1
33.9
29.7
IH
1.27
4.22
8.44
12.7
16.9
21. 1
^53
29.5
33.8
38.0
26.5
I 3/16
1 I.4I
4.70
9.40
14. 1
18.8
23.5
28.2
32.9
37.6
42.3
23.8
iH
: 156
5.21
10.4
15.6
20.8
26.0
31.3
36.5
41.7
46.9
21.5
I 5/16
' 1.72
5.74
1 1.5
17.2
23.0
28.7
344
40.2
459
51.7
19.5
iH
1 1'^
6.30
12.6
1S.9
25.2
31.5
37.8
44.1
50.4
56.7
17.8
I 7/x6
' 2.07
6.89
13.8
20.7
27.6
345
4>3
48.2
55.1
62.0
16.2
1%
1 2.25
7.50
15.0
22.5
30.0
37.5
450
52.5
60.0
67.5
14.9
I 9/x6
, 2.44
8.14
16.3
24.4
32.6
40.7
48.8
57.0
65.1
732
138
IK
< 2.64
8.80
17.6
26.4
35^
44.0
52.8
61.6
70.4
79.2
12.7
I"/i6
2.88
9.60
19.2
28.8
38.4
48.0
57.6
67.2
76.8
86.4
11.7
I^
3.06
10.2
20.4
30.6
40.8
51.0
61.3
71.4
81.6 91.9 1
II.O
I '3/16
' 329
II.
21.9
32.9
43.8
54.8
65.7
76.7
87.6
98.6
10.2
I^
3.52
11.7
234
35.2
46.9
58.6
70.3
82.0
93.8
105.5
§■56
H5/16
: 3.75
12.5
25.0
37.5
50.1
62.6
25 »
87.6
100. 1
112.6
8.95
2
. 4.00
'33
26.7
40.0
53.3
66.7
80.0
93.3
106.7
120.0
8.40
^>^
4.52
15.1
30.1
45.2
60.2
753
90.3
105.4
120.0
135.5
7.43
25<^
5.06
16.9
33.8
50.6
67.1
844
101.3
1 18. 1
135.0
151.9
6.64
2>i
5.64
18.8
37.6
56.4
75.2
94.0
112.8
1316
150.4
169.2
5.96
2>^
6.25
20.8
41.7
62.5
833
10.4
125.0
145.8
166.6
187.5
5.38
*>i
,6.89
23.0
459
68.9
91.9
114.9
137.8
160.8
183.9
206.7
499
2^
• 7.56
25.2
50.4
75.6
100.8
126. 1
151.3
176.5
201.7
226.9
444
2^
S.27
27.6
55.1
82.7
1 10.2
137.8
165.3
192.9
220.4
248.0
4.06
3
9,00
30.0
60.0
90.0
120.0
150.0
180.0
210.0
240.0
270.0
3.73
3X
} 10.6
352
70.4
105.6
140.8
176.0
211. 3
246.5
281.7
316.9
3.17
3>^
1 12.3
40.8
81.7
122.5
163.3
204.2
245.0
285.8
326.7
367.5
2.73
3^
14. 1
46.9
93.8
140.6
187.5
234.4
281.3
328.1
375.0 421.9
2.38
4
1 16.0
53.3
106.7
160.0
2133
266.7
320.0
3730
426.0 1 480.0
2.10
4^
. 18. 1
60.2
120.4
180.6
240.8
301. 1
361.2
421.5
481.7 541.9
1.86
4^
20.3
67.5
1350
202.5
270.0
337.5
405.0
472.5
540.0
607.5
1.66
4¥
1 22.6
Z52
150.4
225.6
300.8
376.1
451.3
526.5
601.7
676.9
1.49
5
25.0
83.3
166.7
250.0
3333
416.7
500.0
583.3
666.7
750.0
».34
^K
27.6
91.9
183.8
275.6
367.5
459.4
551.3
643.1
7350
826.9
1. 21
5^
30.3
100.8
201.7
302.5
403.3
504.2
605.0
705.8
806.7
907.5
I.U
53^
33.1
1 10.2
220.4
330.6
440.8
551.0
661.3
771.5
881.7
991.8
1.02
6
36.0
120.0
240.0
360.0
480.0
600.0
720.0
840.0
960.0
1080
.933
240
WEIGHT OF METALS.
Table No. 76.— WEIGHT OF ROUND IRON.
Lbngth in
Feet.
T _!_
DiAM.
Sect.
Area.
Length
to weig}i
z
a
3
4
5
6
7
8
9
I cwt.
inches.
1
1 sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
feet.
X
1 .0123
.041
.082
.123
.164
.368
.205
.245
.286
.327
.368
2738
3/16
.0276
.092
.184
.276
.460
•552
•644
.736
.828
1217
H
.0491
.164
.327
.491
.655
.818
.982
115
I.3I
1.47
684.4
5/16
.0767
.256
.511
.767
1.02
1.28
1.53
1.79
2.04
2.30
438.1
H
.110
.368
.736
1. 10
147
1.84
2.21
2.58
2.94
3.3>
305.4
7/1 s
.150
.501
I.OO
1.50
2.00
2.51
3.01
35i
4.01
4.51
224.0
>i
.196
.654
1. 21
1.96
2.62
327
3.93
4.58
523
5.89
171.4
9/16
.24S
.828
1.66
2.49
3.31
4.14
4.97
5.80
6.63
7.46
135.5
^.
.307
1.02
2.05
307
4.09
5.II
6.14
7.16
8.66
8.18
9.20
109.5
r
.371
1.24
2.48
3.71
4.9s
6.19
7.42
9.90
II. I
90.6
•442
1.47
2.94
4.42
5.89
7.36 :
8.83
10.3
11.8
13.3
76.0
r
.518
173
346
519
6.91
8.64
10.4
12. 1
13.8
15.6
70.5
.601
2.00
4.01
6.01
8.02
10.0
12.0
14.0
16.0
18.0
559
'5/16
.690
2.30
4.60
6.90
9.20
11.5 1
13.8
16. 1
18.4
20.7
48.7
I
.785
2.62'
' 5.24
7.85
10.5
13." 1
15.7
18.3
20.9
23.6
• 42.8
I ^/i6
.887
2.96
5.91
8.87
11.8
14.8 ,
17.7
20.7
23.6
26.6
37.9
iH
•994
331
6.63
994
^33
16.6
19.9
23.2
26.5
29.8
338
I 3/16
I. II
3.69
7.38
II. I
14.8
18.5
22.2
25.8
29.5
332
30.3
'^,
123
4.09
8.18
12.3
16.4
20.5 '
24.5
28.6
32.7
36.8
27.3
I 5/x6
^•35
45'
9.02
13.5
18.0
22.6
27.1
31.6
36.1.
40.6
24.9
I^
1.48
4.95
9.90
14.9
19.8
24.8
29.7
34.6
39.6
46.6
22.7
I 7/x6
1.62
5.08
10.2
16.2
20.3
25.9
32.5
35. 5
40.6
48.7
20.7
I>i
1.77
5.89
II. 8
17.7
23.6
29.5
35.3
41.2
471
530
19.0
I 9/16
1.92
6.39
12.8
19.2
25.6
32.0 \
38.4
44.7
51.1
57.5
'7.5
ifi
2.07
6.91
13.8
20.7
27.7
34.6,
41.5
48.4
553
62.9
16.2
I"A6
2.24
7.46
14.9
22.4
29.8
37.31
^\
52.2
59.6
67.1
15.0
I^
2.41
8.02
16.0
24.1
32.1
40.1 1
56.1
64.1
72.2
13.9
I » 3/16
2.58
8.60
17.2
25.8
344
43.0
51.6
60.2
68.8
774
13.0
I^
2.76
9.20
18.4
27.6
36.8
46.0
55.2
64.4
73.6
82.8
12.2
1*5/16
2.95
9.83
19.7
29.5
39.3
49.1
59.0
68.8
l^'^o
88.4
11.4
2
3.14
10.5
20.9
31.4
41.9
52.4 j
62.8
733
83.8
94.3
10.7
2>^
3.55
11.8
23.6
355
47.3
59.1 ;
70.9
82.8
94.6
106.4
9.47
2X
3.98
133
26.5
39.8
53.0
66.3
Z25
92.8
106.0
"93
8.44
2^
4.43
14.8
29.5
44.3
59.1
73.8,
88.6
103.3
1 18. 1
132.9
759
2;^
4.91
16.4
32.7
49.1
65.5
81.8,
98.2
1 14. 5
130.9
147.3
6.84
2>i
5.41
18.0
36.1
54.1
72.2
90.2
108.2
126.2
144.3
162.3
6.21
2.V
594
19.8
39.6
59.4
79.2
99.0
118.8
138.5
158.4
178.2
5.66
2J^
6.49
21.6
433
64.9
86.6
108.2
129.8
151.5
1 73. 1
194.8
5.18
3
7.07
23.6
47.1
70.7
94.3
117.8
141.4
164.9
188.5
212.1
4.7s
3^
8.30
27.7
553
83.0
no. 4
138.3 '
165.9
193.6
221.2
248.9
4.05
3>^
9.62
32.1
64.1
96.2
128.3
160.4
192.4
224.5
256.6
288.6
349
3^
II.O
335
23^
1 10.4
147.3
164. 1
220.9
257.7
294.5
33'.3
3.04
4
12.6
41.9
83.8
125.7
167.6
209.4
251.3
293.2
335.0
377.0
2.67
^^
14.2
47.3
94.6
141. 9
189. 1
236.4
283.7
331.0
378.3
425.6
2.37
4H
15.9
53.0
106.0
159.0
212.1
265.1
319. 1
371. 1
424.1
477.1
2.11
aH
17.7
59.1
1 18. 1
177.2
236.3
295.3
354.4
413.5
472.5
531.6
1.90
5
19.6
65.5
130.9
196.4
261.8
327.3
392.7
458.2
523.6
589.1
1.71
5^
21.7
72.2
144.3
216.5
288.6
360.8
432.9
505.1
5773
649.4
1.55
5>^
23.8
79.2
158.4
237.6
316.7
396.0
475.2
554.3
633.6
712.7
I.41
\^
26.0
86.6
173. 1
259.7
346.2
432.8
519.3
605.9
692.4
779.0
1.29
6
28.3
94.2
188.5
282.7
377.0
471.2
565.5
659.7
754.0
848.2
1. 19
ROUND IRON.
241
Weight of Round Iron.
Length in Feet.
T\m a «.*
Sect.
L«n^h
to weigh
^"^*' 1 Arka. I
z
a
3
4
s
6
7
8
9
I ton.
'inches.
sq. in.
332
cwts.
.9876
cwts.
1.975
cwts.
2.963
cwts.
3.950
cwts.
4.938
cwts.
5.926
cwts.
6.613
cwts.
7.901
cwts.
8.888
feet
20.2
6>4
7
38.5
1. 145
2.291
3.436
4.582
5727
6.872
8.018
9.163
10.31
17.5
7}i
44.2
I.315
2.629
3944
5.258
6.573
7.887
9.202
10.52
11.84
15.2
8
503
1.496
2.992
4.448
5.984
7.480
8.976
10.47
11.97
13.46
13.4
8^
56.7
1.689
3.378
5.067
6.756
8.444
10.13
11.82
13.50
15.20
II.8
9
63.6
1.893
3.786
5.680
7572
9.46
11.36
1325
'5i4
17.04
10.6
9^
70.9
2. 1 10
4.220
6.329
8.440
10.55
12.66
14.77
16.88
18.99
9.48
10
78.5
2.338
4.676
7.012
9.352
11.69
1403
16.37
18.70
21.04
8.56
10)^
86.6
2.577
4754
Z7?'
10.31
12.89
15.46
18.04
19.02
2319
776
II
1 950
2.828
5.656
8.485
II. 31
14.14
16.97
19.80
22.62
25.46
7.07
Il>^
; 103 9
3.088
6.176
9.265
12.35
1544
18.53
21.62
24.70
27.80
6.47
1 12
! 1131
3366
6.732
10.10
13.46
16.83
20.20
2356
26.93
30.29
5.94
I2>^
; 122.7
3656
7.312
10.96
14.62
18.28
21.91
2559
29.25
32.90
5.48
13
'132.7
3.950
7.900
tiM
15.80
19.75
23.70
27.65
31.60
35.15
38.34
5.06
i3>^
143. 1
4.260
8.520
12.78
17.04
21.30
25.56
29.82
34.08
4.70
14
153.9
4.581
9.162
13.74
18.32
22.90
26.49
32.07
36.65
41.23
437
14^
1 165.1
4.915
9.830
1474
19.66
2458
28.49
3441
39.32
44.24
4.07
i'5
176.7
5.259
10.52
15.78
21.04
26.30
3146
36.81
42.08
47.33
3.80
15;^
1 188.7
5.616
11.23
16.85
22.46
28.08
32.70
3931
41.89
44.92
50.54
3.56
16
201. 1
5.984
11.97
17.95
23.93
29.92
35.90
47.88
53.86
3.34
16X
213.8
6.364
12.73
19.09
25.46
31.82
38.18
44.55
50.92
57.28
314
17
227.0
6.755
13.51
20.27
27.02
3378
40.53
4729
54.04
60.80
2.96
;I7^
1 240.5
7.»59
14.32
21.48
28.64
355?
42.95
50.11
57.28
6443
2.79
iiS
254.5
2S3.5
Z573
'5i5
22.72
30.29
3786
4544
5301
60.60
68.16
2.64
19
8.438
16.88
25.32
33.75
42.19
50.63
59.03
67.52
75.94
2.37
ao
3142
9350
18.70
28.05
37.40
46.75
56.10
65.45
7480
84.15
2.14
21
346.4
10.31
20.62
30.93
41.23
51.54
^J'li
72.16
82.47
92.78
194
22 I
380.1
II. 31
22.63
3394
45.25
56.57
67.88
79.19
86.56
90.51
101.8
1.77
23 1 4«55
12.37
2473
37.10
49.46
61.83
Z^'g
93.92
III. 3
1.62
24 , 452.4
1
1346
26.93
40.39
53.86
67.32 1
80.78
94.25
107.7
121.3
1.49
16
242
WEIGHT OF METALS.
Table No. 77.— WEIGHT OP ANGLEIRON AND TEEIRON.
I Foot in Length.
Note. — When the base or the web tapers in section, the mean thickness is to be measured.
Thick
NSSS.
Sum of the Width and Depth ik Inches.
inches.
3/16
5/16
I^
lbs.
.81
1.04
1.24
iH
lbs.
.62
.89
I.15
1.37
^H
lbs.
.68
.97
1.25
1.50
^H
lbs.
•73
1.05
1.36
1.63
lbs.
.78
I.I3
1.46
1.76
^>i
lbs.
.83
1. 21
1.56
1.89
2H
^H
lbs.
lbs.
.88
.94
1.29
1.37
1.67
1.77
2.02
2.15
^'A
lbs.
.99
1.45
1.88
2.28
^H
lbs.
1.04
1.52
1.98
2.41
2U
lbs.
1.09
1.60
2.08
2.54
3/i6
S/16
7/16
2%
1. 14
1.68
2.19
2.67
3.13
3.57
1.20
1.76
2.29
2.80
3.28
3.75
3'A
aX
sH
zA
1.25
1.84
2.40
2.93
344
3.93
1.30
1.91
2.50
306
359
4. 1 1
1.45
1.99
2.60
3.19
3.75
4.29
1. 41
2.07
2.71
3.32
391
4.48
ZH
1.46
2. IS
2.81
345
4.06
4.66
zH
1.51
2.23
2.92
3.58
422
4.84
3%
1.56
2.30
3.02
371
4.38
5.02
1.62
2.38
3.13
3.84
4.53
5.20
A%
1.72
2.54
333
4.10
4.84
5.56
3/16
S/x6
H
7/16
91x6
4H
aU
2.70
3.S4
436
5.16
5.92
6.67
7.38
2.85
375
4.62
547
6.29
7.08
7.85
3.01
4.88
5.78
6.65
7.50
8.32
5X
3.16
4.17
5 14
6.09
7.02
.92
.79
I
sH
5H
6X
6>i
3.32
4.38
5.40
6.41
7.38
8.33
9.26
3.48
4.58
5.66
6.72
7.75
8.75
9.73
3.63
479
5.92
7.03
8.11
9.17
10.20
3 79
5.00
6.18
734
8.48
9.58
10.66
3.95
5.21
6.45
7.66
8.84
10.00
II. 13
6^
4.10
5.42
6.71
7.97
9.21
10.42
11.60
4.26
563
6.97
8.28
10.83
12.07
5/16
H
7/16
9/x6
H
1%
583
7.23
8.59
9.93
11.25
12.54
i^8o
rA
7^
6.04
6.25
749
7.75
8.91
9.22
10.30
10.66
11.67
12.08
1301
13.48
14.32
14.84
8
8X
8K
6.46
8.01
9.53
11.03
12.50
13.94
15.36
6.67
8.27
9.84
11.39
12.92
14.41
15.89
6.88
8.53
10.16
11.76
'333
15.88
16.41
8^
9X
^%
7.08
8.79
10.47
12.12
13.75
15.35
16.93
7.39
9.05
10.78
12.49
14.17
15.82
17.45
750
9.31
11.09
12.85
14.58
16.29
17.97
7.71
9.57
II. 41
13.22
15.00
16.76
18.49
9^
7.92
9.83
11.72
13.58
15.42
17.23
19.01
7/16
'A
91x6
zo
loK
12.03
12.66
13.95
14.67
15.83
16.67
17.70
18.63
19.53
20.57
23.13
24.38
ZI
">i
12
13.28
15.40
17.50
19.57
21.61
25.63
13.91
16.13
18.33
20.51
22.66
26.88
14.53
16.86
19.17
21.44
23.70
28.13
ia>^
17.59
20.00
22.38
24.74
29.37
13 13K
14
14^
18.31
20.84
23.31
25.78
30.63
19.04
21.67
24.25
26.83
31.88
19.77
22.50
25.19
27.87
33.13
20.50
2334
26.12
28.91
34.38
15
21.22
24.17
27.06
29.95
3563
12
25.70
28.13
32.45
36.67
izV
24.74
29.37
33.91
38.33
13
13K
25.78
30.63
35.36
40.00
26.83
31.88
36.82
41.67
14
27.87
38.28
4333
15
29.95
3563
41.19
46.67
16
17
18
19
20
32.03
38.13
44.12
50.00
34.12
40.63
47.02
5333
36.20
41.13
49.95
56.67
38.28
43.63
52.87
60.00
40.36
46.13
55.78
63.33
WROUGHTIRON PLATES.
243
Table No. 78.— WEIGHT OF WROUGHTIRON PLATES.
1
Sect.
Akra,
when
I foot
Area in Square Feet.
Number
Thick
mss.
of sq. ft.
in
wide.
I
2
3
4
5
6
7
8
9
X ton.
inch«.
sq. in.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
sq. feet.
X
3.00
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
224.0
5/16
3.75
12.5
25.0
37.5
50.0
62.5
75.0
87.5
100.
112.5
179.2
H
4.50
15.0
30.0
45.0
60.0
75.0
90.0
105.0
120.0
I350
149.3
7/16
5.20
17.5
35.0
52.5
70.0
87.5
105.0
122.5
140.0
157.5
180.0
128.0
^
6.00
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
112.0
9/16
I 6.75
22.5
45.0
67. 5
90.0
112.5 ,
135.0
150.0
180.0
202.5
99.67
H
l^""
25.0
50.0
75.0
lOO.O
125.0 1
150.0
175.0
200.0
225.0
89.60
«»A6
1 8.25
27.5
55.0
82.5
ZIO.O
137.5
165.0
192.5
220.0
247.5
81.45
H
9.00
30.0
60.0
90.0
120.0
150.0
180.0
210.0
240.0
270.0
74.67
»3/i6
9.75
32.5
65.0
97.5
130.0
162.5
195.0
227.5
260.0
292.5
68.92
^
11.50
35.0
70.0
105.0
140.0
175.0
210.0
245.0
280.0
315.0
64.00
^s/x6
11.25
37.5
75.0
112.5
150.0
187.5 '
225.0
262.5
300.0
337.5
5973
X
12.00
40.0
80.0
120.0
160.0
200.0
240.0
280.0
320.0
360.0
56.00
I V«6
12.75
42.5
85.0
127.5
170.0
212.5
255.0
297.5
3+0.0
382.5
52.71
I^
13.50
450
90.0
1350
z8o.o
225.0
270.0
3150
360.0
405.0
49.78
I 3/z6
14,25
47.5
95.0
142.5
190.0
237.5
285.0
332.5
380.0
427.5
47.16
x><
15.0
50.0
lOO.O
150.0
200.0
250.0
300.0
350.0
400.0
450.0
44.80
I^
16.5
550
1 10.0
165.0
220.0
275.0 1
330.0
385.0
440.0
4950
40.73
I^
1 18.0
60.0
120.0
iSo.o
240.0
300.0
360.0
420.0
480.0
540.0
37.33
lF
21.0
1
70.0
140.0
210.0
280.0
350.0
420.0
490.0
560.0
630.0
32.00
2
24.0
80.0
160.0
240.0
320.0
400.0
480.0
560.0
640.0
720.0
28.00
cwts.
cwls.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
2}4
30
.893
1.79
2.68
357
4.46,
§■36
6.25
Z'^
8.04
25.40
3
36
1.07
2.14
3.21
4.29
5.36
6.64
7.50
8.57
9.64
18.67
3>^
^
1.25
2.50
3.75
5.00
6.25
^50
s.57
875
10.00
11.25
16.00
4
48
1.43
2.86
4.29
571
7.14
10.00
"•53
12.86
14.00
4>i
54
1. 61
3.21
4.82
6.43
8.04
9.64
11.25
12.86
14.46
12.44
5
60
1.79
3.57
5.36
7.14
8.93
10.71
12.50
14.29
16.07
11.20
5>^
66
1.96
3.93
5.S9
7.86
9.82
11.79
13.75
15.71
17.68
10.18
6
72
2.14
4.29
6.43
8.57
10.71
12.86
15.00
17.14
19.29
933
7
84
2.50
5.00
Z'5°
10.00
12.50
1500
17.50
20.00
22.50
8.00
8
95
2.86
5.71
8.57
"43
10.29
17.14
20.00
22.86
25.71
7.00
9
108
321
6.43
9.64
12.86
16.07
19.29
22.50
25.71
28.93
6.22
10
120
3.57
7.14
10.71
14.29
12.86
21.43
25.00
28.56
32.14
5.60
Ji
132
393
7.86
11.79
15.71
19.64
2357
27.50
31.43
35.36
5.09
12
144
4.29
8.57
Z2.86
17.14
18.57
21.43
25.71
30.00
34.29
38.57
4.67
J3
'5^
4.64
9.29
13.93
23.21
27.86
32.50
37.14
41.79
431
14
168
5.00
10.00
15.00
20.00
25.00 1
30.00
3500
40.00
45.00
48.21
4.00
•' il
180
5.36
10.71
16.07
21.43
26.79
1
32.14
37.50
42.86
373
244
WEIGHT OF METAI^S.
Table No. 79.— WEIGHT OF SHEET IRON.
AT 480 LBS. PER CUBIC FOOT.
According to Wiregauge used in South Staffordshire (Table No. 17).
Thickness.
Area in Square Fbrt.
B.W.G.
32
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
IS
14
13
12
II
10
9
8
7
6
5
4
3
2
I
inch.
.0125
.0141
.0156
.0172
.0188
.0203
.0219
.0234
.0250
.0281
.0313
•0344
.0375
.0438
.0500
•0563
.0625
.0750
.0875
.1000
.1125
.1250
.1406
.1563
.1719
.1875
.2031
.2188
.2344
.2^00
.2813
.3125
lbs.
.500
.562
.625
.688
.750
.813
.875
.938
1. 00
113
I.2q
1.38
I. SO
I.7S
2.00
2.25
2.50
300
3 SO
4.00
4.50
5.00
5.63
6.25
6.88
7.50
813
8.75
9.38
10.0
11.25
12.5
lbs.
1. 00
I.I3
1.25
1.38
1.50
1.63
1.88
2.00
2.25
2.50
2.75
300
3. SO
4.00
4.50
5.00
6.00
7.00
8.00
9.00
lO.O
"3
12.5
13.8
15.0
16.3
17.S
18.8
20.0
22.5
25.0
lbs.
1.50
1.69
1.88
2.06
2.25
2.44
2.63
2.81
3.00
3.38
3.7s
4.13
4.50
5.25
6.00
6.75
7. SO
9.00
10.5
12.0
13.5.
15.0
16.9
16.8
20.6
22.5
24.4
26.3
28.1
30.0
33.8
37.5
lbs.
2.00
2.25
2.50
2.75
3.00
3.2s
3.50
3.7s
4.00
4.50
5.00
5.50
6.00
7.00
8.00
9.00
10.0
12.0
14.0
16.0
18.0
20.0
22.5
25.0
275
30.0
32. S
3S.O
37. S
40.0
45.0
50.0
lbs.
2.^0
2.81
313
344
375
4.06
4.38
4.69
Soo
S.63
6.25
6.88
7.50
8.75
10. o
"•3
12.5
150
17.5
20.0
22.5
2q.O
25.1
313
34.4
37.S
40.6
43.8
46.9
50.0
56.3
62.S
lbs.
3.00
3.38
3.75
4.13
4 JO
4.88
5.2s
S.63
6.00
6.75
7.50
8.25
9.00
10.5
12.0
13.5
IJ.O
18.0
21.0
24.0
27.0
30.0
33.8
37. S
41.3
45.0
48.8
S2.S
56.3
60.0
67.5
75.0
lbs.
350
3.94
4.38
4.81
52S
5.69
6.13
6.56
7.00
7.88
8.7s
9.63
10.5
12.3
14.0
15.8
I7.S
21.0
24. S
28.0
31.5
3S.O
494
43.8
48.1
S2.5
56.9
61.3
65.6
70.0
78.8
87. S
8
lbs.
4.00
4.50
5.00
SSo
6.00
6.50
7.00
7.50
8.00
9.00
lO.O
II.O
12.0
14.0
16.0
18.0
20.0
24.0
28.0
32.0
36.0
40.0
45.0
50.0
SS.o
60.0
65.0
70.0
7S.0
80.0
90.0
100.0
lbs.
4.50
5.06
5.63
6.19
6.75
7.31
7.88
8.44
9.00
10. 1
"3
12.4
13.S
18.0
20.3
22.5
27.0
31.S
36.0
40.5
45.0
50.6
S6.3
61.9
67.5
72.1
78.8
84.4
90.0
101.3
112.5
Number
of sq. ft.
in I ton.
sq. ft
4480
3986
3584
32S6
2987
275s
2560
2388
2240
1982
1792
1623
1493
1280
1 120
996
896
747
640
560
498
448
398
358
326
299
276
256
239
224
199
179
IRON SHEETS.
24s
Table No. 80.— WEIGHT OF BLACK AND GALVANIZED
IRON SHEETS.
(Morton's Table, founded upon Sir Joseph Whitworth & Co.*s Standard
Birmingham WireGauge.)
XoTE. — ^The numbers on Holtzapflfel's wiregauge are applied to the thicknesses
on Whitworth's gauge.
Gauge of Black Sheets.
Approximate number of
sqtiare feet in i ton.
Gauge of Black Sheets.
Approximate ntmiber of
square feet in x ton.
Wire
Thickness.
Black
Galvanized
Wire
Thickness.
Black
Galvanized
Gauge.
Sheets.
Sheets.
Gauge.
Sheets.
Sheets.
No.
inch.
square feet
square feet.
No.
inch.
square feet
square feet
I
.300
187
185
17
.060
933
876
2
.280
200
197
18
.050
1120
1038
3
.260
215
212
19
.040
1400
1274
4
.240
233
229
20
.036
1556
1403
5
.220
^§^
250
21
.032
1750
1558
6
.200
2^^
275
22
.028
2000
1753
7
.180
3"
304
23
.024
2333
2004
8
.165
339
331
24
.022
2545
2159
9
.150
373
363
^§
.020
2800
2339
10
.135
415
403
26
.0x8
3111
?^
II
.120
467
452
H
.016
3500
12
.110
509
491
28
.014
4000
3122
'3
.095
589
566
29
.013
4308
3306
14
.085
659
630
30
.012
4667
3513
15
.070
800
757
31
.010
5600
4017
16
1
.065
862
813
32
.009
6222
4327
246
WEIGHT OF METALS.
Table No. 81.— WEIGHT OF HOOP IRON.
I FOOT IN LENGTH.
According to Wiregauge used in South Staffordshire.
Thicj
Width in
Inches.
CNBSS.
H
^
H
I
^}i
IX
iH
i>4
B.W.G.
inches.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs. 1
lbs.
21
20
19
■0344
.0375
.0438
.0716
.0781
.0911
.0861
.0938
.109
.100
.109
.128
.115
.146
.129
.141
.164
.144
.156
.182
.158
.172
.200
.172
.188
.219
18
16
.0500
.0563
.0625
.104
.117
.130
.125
.141
.156
.146
.164
.182
.167
.188
.208
.188
.211
.234
.208
.234
.260
.229
.258
.286
.250
.281
.313
15
13
.0750
.0875
.1000
.156
!208
.188
.219
.250
.219
.256
.292
.250
.293
.333
.281
.329
.375
•313
.366
.416
.344
.402
•458
.375
.438
.500
12
II
10
.1125
.1250
.I40&
.234
.260
.293
.281
.313
.352
.328
•365
.410
.375
.417
.469
.422
.469
.527
.469
.521
.586
.516
•573
.645
.563
.625
.703
7
.1563
.1719
.1875
.326
.358
.391
.391
.430
.469
.456
.501
.547
.522
.573
.625
•587
.644
•703
.652
.716
.781
.717
.788
.859
•783
.859
.938
6
5
4
.2031
.2188
.234+
■423
.488
.508
:r8^
.030 •
.683
.677
.729
.781
.762
.820
.879
.836
.912
.977
.931
10.0
10.7
1.02
1.09
I.I7
KNBSS.
Width in
Inches.
IHIC
^H
Il<
I^
a
2X
2>i
2J<
3
B.W.G.
inches.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
21
20
19
.0344
.0375
.0438
.197
.203
.238
.201
.219
.257
.215
.224
.274
.229
.250
.292
.258
.251
.328
.287
.313
.365
•315
•344
.400
.344
•375
.437
18
17
16
.0500
.0563
.0625
.271
.305
•339
.292
.328
.365
.312
.351
.391
•333
•375
.417
.375
.422
.469
.417
.469
.521
.458
.516
.573
.500
.563
.625
15
13
.0750
.0875
.1000
.307
.475
•543
.438
.584
.469
•549
.626
.585
.667
.562
.658
.750
.625
.833
.687
.804
.917
.750
.875
I.OO
12
II
10
.1125
.1250
.1406
.609
.677
.762
.656
.729
.820
.703
.781
.879
.750
•833
.938
.842
.937
1.06
.938
1.04
1.17
1.03
1. 15
1.29
I.I3
1.16
7
.1563
.1719
.1875
.848
.931
1.02
•913
1. 00
1.09
.978
1.07
1.17
1.04
i.iS
1.25
I.I7
1.29
1.41
1.30
1^43
1.56
1.58
1.72
1.56
1.72
1.88
6
5
4
.2031
.2188
.2344
1. 10
1.19
1.27
1.28
1.37
1.27
1.37
1.46
135
1.46
1.56
1.64
1.76
1.69
1.82
i'9S
1.86
2.C0
2.15
2.03
2.19
2.35
WARRINGTON IRON WIRE.
247
Table No. 82.— WEIGHT AND STRENGTH OF WARRINGTON
IRON WIRE.
Table of Wire manufactured by Rylands Brothers.
Note.— The WireGauge is that of Rylands Brothers.
1
i
1
Specific
1
Size on ;
Weight of II
Length of j
Breaking Strain. 
Density,
Wire 1
Gauge.
Diameter.
100 Yds.
iMUe.
T Bundle
of 63 lbs.
I Cwt.
1
An
nealed.
Bright.
the aver
age den
sity of iron
=1.
inch.
milli
metres.
lbs.
lbs.
yards.
yards. ,
lbs.
lbs.
average
iron
= I.
7/d
H
12.7
193.4
3404
33
il
10470
15700
I.OI68
«/o
^Vsa
II.9
170.0
2991
37
66
9200
I381O
5/0
7/16
II.I
148. 1
2606
43
It
8020
12000
4/0
'3/3a
10.3
127.6
2247
49
88 1
6910
10370
3/0
H
n
108.8
I915
r
103
5890
8835
I.O168
Vo
"/3a
91.4
1609
69
123
4960
7420
.326
8.3
82.1
1447
77
136
4450
6678
I
.3a>
7.6
69.6
1227
^
161
3770
5655
2
.274
7.0
5?^
1022
108 '
193
3140
4717
3
.250 (1)
6.4
48.4
851
130
232
2618
3927
I.OI68
4
.229
5.8
40.6
714
*55
276 1
2197
3295
5
.209
5.3
33.8
595
186
332
1830
2740
6
.191
4.9
28.2
495
223
397
1528
2290
7
.174
4.4
23.4
412
269
479
1268
1900
8
•159
4.0
19.6
344
322
573
1060
1558
9
.146
3.7
16.5
290
3^
680
893
1340
10
•133
34
13.7
241
460
819
741
I no
10^
.125 (i)
3.2
12. 1
213
521
927
654
980
I.OI79
II
.117
30
10.6
186
595
1059
573
860
12
.100 (A)
2.6
8.0
142
783
1393
436
650
13
.090
2.3
n
no
1006
1790
339
509
14
.079
2.0
!5
1305
2322
261
390
IS
.069
1.8
3.7
65
1715
2188
3052
199
299
16
.0625(A)
1.5
2.9
51
3894
156
233
1.0690
17
•053
13
2.2
38
2900
5160
118
176
18
.047
1.2
1.7
30
3687
6560
93
138
19
.041
I.O
1.3
23
4847
8620
70
^25
20
.036
.9
1.0
18
5985
11120
54
81
21
.03125(A)
.8
.8
H
7574
14152
43
64
1. 1765
22
.028
.7
.6
II
9893
18486
33
49
A/^m. This Table of the weight and strength of Warrington wire is given by permission
of Messrs. Rylands Brothexs; and it is said to be based on very accurate measurements of
sires and weights. The last column is added by the author, to show that the density of
the wire is stationary for diameters of from yi inch to X inch, and probably somewhat
smaller diameters; but that, contrary to current opinions of the density of wire, the
density becomes greater when the diameter is reduced to }i inch, and is gradually
increased as the diameter is further reduced.
248 WEIGHT OF METALS.
Table No. 83.— WEIGHT OF WROUGHTIRON TUBES,
Bv Internal Diameter.
Length, i Foot, Thickness by Holuapffel's WireGauBe
WROUGHTIRON TUBES.
249
Table No. 83 (continued).
Length, i Foot. Thickness by Holtzapffers WireGauge.
Thick
)
MCSS.
W. G.
8
9
10
II
12
13
14
15
16
17
18
Inch.
.165
.148
•134
.120
.109
.095
.083
.072
.065
.058
.049
"/64^
9/64/
9/64 b.
%b.
7/64
3/32/
5/64/
s/64^.
V16/
Vi6^.
3/64/
Int.
DiAM.
lbs.
lbs.
lbs.
lbs.
lbs.
1
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
inches.
H
.501
.423
.364
•3*8
.267
.219
.181
.149
.130
.Ill
.0895
%
.717
.610
•539
.472
.410
•343
.290
.243
.215
.187
•»54
H
.934
.797
.714
.625
.553
.468
.398
.337
.300
.263
.218
%
1.15
1. 00
.890
■779
.695
.592
.507
•431
.385
.339
.282
H
1.58
139
1.24
1.09
.981
.841
.718
.620
•555
.491
.410
1
2.01
1.78
159
1.41
1.27
1.09
.935
.808
•Z^5
.643
.538
iH
2.45
2.17
1.94
1.72
1.55
1.34
I.I5
.997
.895
.795
.667
^%
2.88
2.55
2.29
2.04
1.84
1.59
137
1. 19
1.07
.946
.795
^M
33'
2.94
2.64
2.35
2.12
1.84
'•59
1.37
1.24
1. 10
.923
2
3.74
333
3.00
2.66
2.41
2.08
1. 81
1.56
1.41
1.25
1.05
2%
4.17
3.72
335
2.98
2.69
2.33
2.02
1.75
1.58
1.40
1. 18
2%
4.61
4.10
3.70
329
2.98
2.58
2.24
1.94
1.75
1.55
1.31
2^
504
^2S
4.05
3.61
3.26
2.83
2.46
2.13
1.92
Hi
1.44
3 ^
5.47
4.88
4.40
392
3.55
3.08
2.68
2.31
2.09
1.86
'•57
Z}i
6.33
5.65
5.10
^•55
4.12
3.58
3"
2.69
2.43
2.16
1.82
4
7.20
6.43
5.80
5.18
4.69
4.07
3.55
307
2.77
2.47
2.08
4K
8.06
7.20
6.50
5.81
5.26
4.57
3.98
345
3."
2.77
'2.34
5
8.93
7.98
7.21
6.44
5.83
5.07
4.42
3.83
3.45
3.07
2.59
\^
9.79
8.75
7.91
7.06
6.40
5.57
4.85
4.20
3.79
3!
2.85
6
10.7
9.53
8.61
7.69
6.97
6.07
5.29
4.58
4. 13
3.68
3. 1 1
1 6j^
11.5
10.3
9.31
8.32
7.55
6.56
§•72
4.96
4.47
3.98
3.36
' 7
12.4
ii.i
10.
8.95
8.12
7.06
6.16
533
4.81
4.29
3.62
l^
133
II. 9
10.7
958
8.69
Z55
6.59
5.71
5.15
4.59
3.88
8
14. 1
12.6
11.4
10.2
9.26
8.06
7.03
6.09
549
4.90
4.13
9
15.8
14.2
12.8
II. 5
10.4
9.05
7.90
6.84
6.17
5.50
4.65
10
17.6
»5.7
14.2
12.7
II.5
lO.O
8.77
7.60
6.85
6.11
5.16
II
19.3
^73
15.6
14.0
12.7
II.
964
8.35
753
6.72
5.67
12
21.0
18.8
17.0
15.2
13.8
12.0
10.5
9.10
8.21
7.33
6.19
13
22,7
20.4
18.4
16.5
15.0
130
11.4
9.86
8.89
793
6.70
14
245
21.9
19.8
17.7
16. 1
14.0
12.2
10.6
9.57
8.54
7.22
'5
26.2
235
21.3
19.0
17.2
15.0
13.1
11.4
10.3
Kl
7.73
16
1
27.9
25.0
22.7
20.3
18.4
16.0
14.0
12. 1
10.9
9.88
8.24
I '7
29.6
26.6
24.1
21.5
19.5
17.0
14.9
12.9
11.6
10.4
8.76
' 18
*
3»4
28,1
255
22.8
20.6
; 18.0
15.7
13.6
12.3
II.
9.27
;i9
33.1
29.7
26.9
24.0
21.8
1 19.0
16.6
14.4
130
11.6
9.78
• 20
348
31.2
28.3
253
22.9
20.0
^Z5
15.1
13.7
12.2
10.3
10.8
21
36.6
32.8
29.7
26.?
24.1
21.0
18.3
^H
14.3
12.8
22
38.3
34.3
311
27.8
25.2
1 22.0
19.2
16.6
15.0
13.4
II. 3
23
40.0
35.9
32.5
29.1
26.4
1 23.0
20.1
17.4
15.7
14.0
II. 8
24
41.8
37.4
339
303
27.5
1 24.0
20.9
18. 1
16.4
14.6
12.6
26
45.2
48.7
40.5
367
32.8
29.8
1 26.0
22.6
197
17.7
15.8
134
28
436
39.5
35.3
32.1
28.0
24.4
21.2
19. 1
17.0
14.4
30
52.1
46.7
42.3
37.8
34.4
30.0
26.1
22.7
20.5
18.3
15.4
32
55.5
49.8
48.0
40.4
36.7
32.0
27.9
24.2
21.8
19.5
16.5
34
59.0
52.9
42.9
39.0
34.0
29,7
25.8
23.2
20.7
^I'i
>36
62.4
56.0
50.8
454
41.3
1 36.0
314
27.3
24.6
21.9
18.6
250
WEIGHT OF METALS.
Table No. 84.— WEIGHT OF WROUGHTIRON TUBES,
BY External Diameter.
Length, i Foot. Thickness by Holtzapffel's Wire Gauge.
Thickness.
W. G.
7
8
9
ID
II
la
13
14
15
Inch.
.180
.165
.148
.134
.120
.109
.095
.083
.072
3/x6 ^.
"/64 ^•
9/64/
9/64 ^.
yib
7/64
3/33/
5/64/
5/64 b.
Ext. Diam.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
I inch.
'•55
1.44
132
1.22
I. II
1.02
.900
.797
.700
'H
1.78
1.66
I.5I
139
1.26
1. 16
1.03
.906
•794
'^
2.02
1.88
I.7I
I.S7
1.42
1.30
1. 15 1 I.OI
.888
'H
2.25
2.09
1.90
1.74
1.58
1.45
1.27 ! 1. 12
.983
'^
2.49
2.31
2.10
1.92
^'P
1.59
1.40 ! 1.23
1.08
iH
2.72
2.52
2.29
2.09
1.89
H'^
1.52 1 1.34
1.17
'H
2.96
2.74
2.48
2.27
2.05
1.87
1.65 , 1.45
1.27
m
3.19
2.96
2.68
2.45
2.21
2.02
1.77 I 1.56
1.36
2
343
3.17
2.87
2.62
2.36
2.16
1.90 1 1.67
145
i%
3.67
339
3.06
2.80
2.52
2.30
2.02
1.78
155
^%
3.90
3.60
326
2.97
2.68
2.44
2.14
1.88
1.64
2H
4.14
3.82
3.45
3.15
2.83
2.59
2.27
1.99
1.74
*^
4.37
4.04 3.65
332
2.99
2.73
2.39
2.10
1.83
2H
4.61
4.25
3.84
350
3.15
2.87
2.52
2.21
193
2¥
4.84
4.47
4.03
3.67
331
3.02
2.64
2.32
2.02
2^
5.08
4.68
4.23
3.85
346
3.16
2.77
2.43
2.11
3 ,
5.32
4.90
4.42
4.02
3.62
3.30
2.89
2.54
2.21
yx
VI
5.33
4.81
437
394
359
314
2.75
2.40
i'A
6.26
5.76
5.20
4.72
4.25
387
3.39
2.97
2.59
2K
6.73
6.19
5.58
507
4.88
4.16
3.64
3.19
2.77
4
7.20
6.63
5.97
5.43
4.44
3.89
340
2.96
A%
7.67
7.06 6.36
5.78
5.20
4.73
4.38
3.62
3.15
*>i
8.14
749
7.45
6.13
55'
5.01
3.84
334
4^
8.61
7.91
713
6.48
5.82
530
4.63
4.06
353
s ,
9.08
8.35
7.52
6.83
6.13
5.58
4.88
4.27
372
5^
9.56
8.79
7.91
8.30
7.18
6.44
5»7
5.13
4.49
390
s'4
10.0
9.22
7.88
6.76
6.15
538
4.71
4.09
k^
10. 5
9.65
8.68
7.07
6.44
5.63
4.93
4.28
6
II.O
ZO.I
9.07
8.23
7.39
6.73
5.87
5.14
5.36
4.47
6H
11.4
10.5
9.46
8.58
7.70
8.02
7.01
6.12
4.66
(,'A
11.9
10.9
9.85
8.93
7.30
6.37 558
4.8s
(>K
12.4
11.4
10.2
9.28
8.33
7.58
6.62
579
503
7
12.9
1 1.8 10.6
9.63
8.64
7.87
6.87
6.01
5.22
1%
'33
12.2 II.O
9.99
8.96
8.15
7.12
6.23
541
1%
13.8
12.7
11.4
10.3
9.27
8.44
7.37
6.45
5.60
l^
14.3
131
II. 8
10.7
959
8.72
7.62
6.66
579
8
14.7
13.5
12.2
II.O
9.90
9.01
7.86
6.88
5.98
Thicknkss.
W. G.
4
5
6
7
8
9
Inch.
•3125
.281
.238
.220
.2
J03
*
180
.165
.148
5/16
9/3a
^5/64/
7/33
r X3
'/64
^
Ueb.
"/64 ^
9/64/
Ext. Diam.
lbs.
lbs.
lbs.
lbs.
"7
bs.
1
lbs.
lbs.
lbs.
7 inch.
21.9
19.8
16.9
15.6
li
^5
2.9
II. 8
10.6
l^
235
21.3
18. 1
16.8
II
>•§
3.8
12.7
II.4
8
25.2
22.7
19.3
17.9
i(
3.6
4.7
'35
12.2
8^
26.8
24.2
20.6
19. 1
i:
r.6
57
14.4
12.9
9
28.4
257
21.8
20.2
i\
H
6.6
15.3
137
9>^
30.1
27.1 ! 23.1
21.4
i<
?.8
7.6
16. 1
14.5
10
31.7
28.6 ! 24.3
22.5
1 ^
>.8
8.5
17.0
15.3
LIST OF TABLES OF CAST IRON, STEEL, ETC. 2$ I
LIST OF TABLES OF THE WEIGHT OF CAST IRON,
STEEL, COPPER, BRASS, TIN, LEAD, AND ZINC.
The following Tables are devoted to the specialities of manufacture in
Cast Iron, Steel, and other metals, embracing the utmost range of dimen
sions to which objects in the several metals are executed in the ordinary
course of practice.
Thus, whilst it is customary for certain classes of Cylinders in Cast Iron —
steam cylinders, for example — to be constructed according to given internal
diameters, other classes are constructed according to diameters given
externally, as the iron piers of railway bridges. Two distinct tables accord
ingly have been composed, showing the weights of Cylinders of various
thicknesses, and of diameters as measured internally and externally.
The weights of Copper Pipes and Cylinders are only calculated for in
ternal diameters, as it is not the practice to construct them to given external
diameters. Brass Tubes, on the contrary, are calculated only for external
diameters, as they are not ordinarily made to given internal diameters.
Table No. 85. — ^Weight of Castiron Cylinders, i foot in length, advanc
ing, by internal measurement, from i inch to 10 feet in diameter, and from
}^ inch to 2j^ inches in thickness.
Table No. 86. — ^Weight of Castiron Cylinders, i foot in length, advanc
ing, by external measurement, from 3 inches to 20 feet in diameter, and
from ^Ixt inch to 4 inches in thickness.
Table No. 87. — Volume and weight of Castiron Balls, when the
diameter is given; from i inch to 32 inches in diameter, with multipliers
for other metals.
Table No. 88. — Diameter of Castiron Balls, when the weight is given ;
from yi pound to 40 cwts.
Table No. 89. — ^Weight of Flat Bar Steel, i foot in length ; from ^ inch
to I inch thick, and from }i inch to 8 inches in width.
Table No. 90. — Weight of Square Steel, i foot in length ; from }i inch
to 6 inches square.
Table No. 91. — ^Weight of Round Steel, 1 foot in length; from }i inch
to 24 inches in diameter.
Table No. 92. — ^Weight of Chisel Steel: hexagonal and octagonal, i foot
in length; from ^ inch to i^ inches diameter across the sides.
Ovalflat, from J^ x ^ inch to ij5^ x ^ inch.
Table No. 93. — ^Weight of one square foot of Sheet Copper; from No. i
to No. 30 wiregauge, as employed by Williams, Foster, & Co.
Table No. 94. — Weight of Copper Pipes and Cylinders, i foot in length,
advancing, l)y internal measurement, from }i inch to 36 inches in diameter,
and from No. 0000 to No. 20 wiregauge in thickness.
252 WEIGHT OF METALS.
Table No. 95. — ^Weight of Brass Tubes, i foot in length, advancing, by
external measurement, from }i inch to 6 inches in diameter, and from
No. 3 to No. 25 wiregauge in thickness.
Table No. 96. — ^Weight of one square foot of Sheet Brass; from No. 3 to
No. 25 wiregauge in thickness.
Table No. 97. — Size and weight of Tin Plates.
Table No. 98. — ^Weight of Tin Pipes, as manufactured.
Table No. 99. — Weight of Lead Pipes, as manufactured.
Table No. 100. — Dimensions and weight of Sheet Zinc. (VicUeMon
tagne,)
CASTIRON CYLINDERS.
253
Table No. 85. — Weight of CastJron Cylinders.
By Internal Diameter, i Foot Long.
Int.
Thickness in Inches.
DiAM.
X
5/16
H
7/16
>i
9/16
H
"/I6
H
H
I
inches.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
I
307
4.03
5.06
6.17
7.36
8.63
9.97
II.4
12.9
16. 1
19.6
^'A
4.30
5.56
6.90
8.32
9.82
II.4
13. 1
14.8
16.6
20.4
245
2
5.52
7.09
8.74
10.5
12.3
14.2
16. 1
18. 1
20.3
247
29.5
2>i
6.75
8.63
10.6
12.6
147
16.9
19.2
21.5
239
29.0
344
3 ,
7.9«
10.2
12.4
14.8
17.2
19.7
22.2
24.9
27.6
33.3
39.3
3>i
9.20
II.7
143
16.9
19.6
22.4
25.3
28.3
3'.3
37.6
44.2
4 ^
10.4
13.2
16.1
19. 1
22.1
2Q.2
28.0
28.4
31.6
350
41.9
49.1
4^
II.7
14.8
18.0
22.1
245
31.5
350
38.7
46.2
54.0
5 ,
12.9
16.5
19.8
23.4
27.0
307
345
38.4
42.3
50.5
1^2
5^
141
17.8
21.6
255
29.5
33.5
37.6
41.8
46.0
54.8
ti'^
^
15.3
19.4
235
27.7
32.0
36.2
40.7
45.1
49.7
59.1
68.7
Thickness in Inches.
1
H
7/x6
H
9/x6
H
"/»6
H
H
I
IH
iH
inches.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lU.
lbs.
6
23.5
27.7
320
36.2
40.7
45.1
49.7
591
68.7
78.7
89.0
^H
25.3
29.8
344
390
^37
48.5
53.4
63.4
73.6
84.2
951
7 ,
27.2
32.0
36.8
41.8
46.8
51.9
57;
67.7
89.7
101.2
7}i
29.0
34.1
39.3
44.5
49.9
55.3
60.8
71.9
95.3
107.4
8
30.8
36.3
41.7
47.3
52.9
58.6
64.4
76.2
80.5
100.8
"35
8)^
32.7
38.4
44.2
50.0
55.9
62.0
68.1
933
106.3
119. 7
' ^.
34.5
40.5
46.6
52.8
59.0
U.t
71.8
84.8
98.2
III.8
125.8
. 9>^
36.4
42.7
49.1
55.6
58.3
61. 1
62.0
75.5
89.1
lOJI
108.0
112.9
117.4
131.9
• 10
1 «o>^
3».2
40.0
44.8
47.0
51.5
54.0
65.1
68.2
72.1
755
82.8
93.4
97.7
122.9
128.4
138. 1
144.2
' 11
41.9
49.1
56.5
63.9
71.2
78.9
86.5
102.0
1 17.8
133.9
150.3
11;^
437
51.3
58.9
66.6
745
82.3
90.2
106.3
122.7
139.4
156.5
12
45.6
53.4
61.4
69.4
83.6
85.6
939
no. 6
127.6
145.0
162.6
1 13
49.2
57.7
66.3
749
80.4
92.4
101.2
1 19.2
137.5
156.0
1749
187.2
. ^^
52.9
62.0
71.2
89.7
99.1
108.6
127.8
147.3
167. 1
178.1
1 15
^1
66.3
76.1
85.9
959
105.9
116.0
136.4
157. 1
199.4
16
70.6
81.0
91.5
102.0
112.6
123.3
145.0
166.9
189. 1
211.7
17
64.0
749
85.9
97.0
108.2
1 19. 4
130.7
1536
176.7
186.5
200.2
224.0
18
67.7
79.2
9o.8
102.5
"43
126. 1
138. 1
162.2
211. 2
236.2
Thickness in Inches.
H
7/16
H
H
H
H
I
iH
iX
m
i^
inches.
cwt.
cwt.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
' 18
.604
.707
.811
1.02
1.23
145
1.67
1.89
2.II
2.34
2.46
2.56
19
.637
.746
:S
1.08
1.30
1.52
1.75
1.99
2.22
2.70
20
.670
.784
113
1.36
1.60
1.84
2.08
2.33
2.58
2.83
21
.703
.823
.942
1. 19
143
1.68
1.93
2.18
2.44
2.70
2.96
22
.736
.861
.986
1.24
1.49
1.76
2.02
2.28
2.55
2.82
309
' 23
.769
.900
1.03
1.29
1.56
1.83
2.10
2.38
2.66
2.94
322
24
.802
.939
1.07
1.35
1.63
1.91
2.19
2.48
HI
3.06
335
25
.835
.977
1. 12
1.40
1.69
1.99
2.28
2.58
2.88
3.18
348
254
WEIGHT OF METALS.
Table No. 85 {continued).
By Internal Diameter, i Foot Long.
Int.
Thickness in
Inches.
DiAM.
H
7/x6
%
H
H
n
I
1%
1%
m
i}i
inches.
CWU.
cwts.
cwts.
cwts.
t
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
26
.868
1.02
1. 16
1.46
1.76
2.06
2.37
2.68
2.99
330
362
27
.901
1.05
I.2I
I.5I
1.82
2.14
2.45
2.77
309
3.42
HI
28
.934
1.09
1.25
157
1.89
2.22
2.54
2.87
3.20
3.54
3.8S
29
.967
i.i:«
1.29
1.62
1.96
2.29
2.63
2.97
3.31
3.66
4.01
30
.998
1.17
1.34
1.68
2.02
2.37
2.72
3.07
342
378
4.14
32
1.06
1.25
143
1.79
2.15
2.52
2.89
327
364
4.02
4.41
3i
1.13
1.32
1.51
1.90
2.29
2.67
3.07
346
3.86
4.26
4.67
36
1.20
1.40
1.60
2.01
2.42
2.83
3.24
3.66
4.08
450
4.94
38
1.26
1.47
1.69
2.12
^•§§
2.98
3.42
3.86
4.30
4.75
5.20
40
1.33
1.55
1.77
2.23
2.68
314
3.59
4.05
4.52
4.99
5.47
42
1.39
1.63
1.86
2.34
2.81
3.29
3.77
4.25
4.74
523
5.73
45
1.49
1.75
1.99
2.50
3.01
3.52
4.03
4.55
5.07
55?
6.13
48
1.59
1.86
2.12
2.66
321
375
430
4.85
5.40
5.96
6.52
Thick
NBSS IN
Inches.
H
H
Ji
1
>X
1%
iji
i>i
I^
2
2X
inches.
cwts.
cwts*
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
48
2.66
3.21
375
430
4.85
5.40
5.96
6.52
7.63
8.77
9.91
SI
2.82
3.40
3.98
4.56
S'H
573
6.32
6.91
8.09
9.29
10.5
54
2.99
3.60
4.21
4.82
544
6.06
6.69
7.31
8.55
9.82
II.I
57
3.15
3.80
444
5.09
5.73
6.03
6.38
7.05
7.70
8.10
9.01
10.4
II.7
60
3.32
4.00
4.67
535
6.71
7.41
947
10.9
12.3
63
3.48
4.19
4.90
5.61
6.33
7.04
7.78
8.49
993
II.4
12.9
66
3.64
439
5.13
5.88
6.62
7.37
8.14
8.89
10.4
II.9
135
69
3.81
4.59
5.36
6.14
6.92
7.70
8.51
9.28
10.9
12.5
14.1
72
3.97
4.78
5 §9
6.40
7.21
8.03
8.87
9.67
"•?
130
14.7
75
4.14
4.98
5.82
6.66
75'
8.36
9.24
10. 1
11.8
13.5
15.2
78
4.30
5^!
6.05
6.93
7.81
8.69
9.60
10.5
12.2
14.0
15.8
81
4.46
5.38
6.28
719,
8.10
9.02
9.97
10.9
12.7
14.6
16.4
84
4.63
557
6.51
7.45*
8.40
9.35
10.3
"3
13.2
I5.I
17.0
87
4.79
5.77
6.74
7.72
8.69
9.67
10.7
II. 6
136
15.6
17.6
90
4.96
597
6.97
7.98
S.24
8.99
10.0
II.I
12.0
14. 1
16. 1
18.2
93
5.12
6.17
7.20
9.29
10.3
1 1.4
12.4
14.5
16.7
18.8
96
5.28
6.36
HI
8.51
9.58
9.88
10.7
II. 8
12.8
15.0
17.2
19.4
99
545
6.56
7.66
8.77
II.O
12.2
13.2
15.5
177
20.0
102
5^i
6.76
7.89
903
10.2
"3
12.5
136
15.9
18.2
20.6
loq
5.78
6.95
8.12
9.29
10.5
II. 7
12.9
14.0
16.4
18.8
21.2
108
5.94
7.15
8.36
9.56
10.8
12.0
133
14.4
16.8
'93
21.8
III
6.10
7.35
fS9
9.82
ii.i
12.3
136
14.8
17.8
19.8
22.3
114
6.27
7.55
8.82
10. 1
11.4
12.6
14.0
15.2
20.3
22.9
117
6.43
7.74
9.05
10.4
11.7
13.0
14.3
K.6
iS.o
18.2
20.9
235
120
6.59
794
9.28
10.6
12.0
13.3
14.7
18.7
21.4
24.1
CASTIRON CYLINDERS.
^SS
Table No. 86. — Weight of CastIron Cylinders.
By External Diameter, i Foot Long.
Ext.
Thickness in
Inches.
DiAM.
1
3/16
X
5/x6 '
I
H
7/16
K
9/16
H
U
H
I
inches.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
3
5.18
6.75
7.98
8.25,
9.65
II.O
12.3
13.5
14.6
16.6
18.3
19.6
Z}^
6.10
9.78!
11.5
13.2
14.7
16.2
17.6
20.3
22.6
24.5
4 ,
7.02
9.20
"3
13.3
15.3
17.2
19.0
20.7
23.8
24.0
26.9
29.5
4,^
794
10.4
12.9
15.2
17.5
19.6
21.7
27.7
31.1
344
5 ,
8.86
II.7
14.4
17.0
19.6
22.1
24.5
26.9
31.5
35.4
39.3
5^
9.78
12.9
15.9
18.9
21.8
24.5
27.3
29.9
352
39.7
44.2
6
10.7
14. 1
17.5
20.7
23.9
27.0
30.0
33.0
38.9
44.0
49.1
6X
II. 6
*$•?
19.0
22.5
26.0
29.5
32.8
36.1
42.6
48.3
54.0
7
12.5
16.6
20.5
24.4
28.2
31.9
356
39.1
46.4
52.6
58.9
l}i
13.5
17.8
22.1
26.2
30.3
34.4
38.3
42.2
5°i
56.9
63.8
68.7
8
14.4
19.0
23.6
28.1
32.5
36.8
41.1
^§•3
538
61.2
8^
153
20.3
251
29.9
34.6
393
438
48.3
57.5
^J'i
73.6
9
16.2
21.5
26.7
31.8
36.8
41.7
46.6
51.4
61.3
69.8
9H
lo
17.2
18.1
22.7
23.9
28. 2
29.7
33.6
354
38.9
41. 1
44.2
46.6
49.4
52.1
54.5
57.5
65.0
68.7
78.4
II
19.9
26.4
32.8
39.1
454
51.5
57.6
63.7
76.0
87.0
98.2
12
21.8
28.8
35.9
42.8
49.7
56.5
63.2
69.8
83.4
95.6
108.0
13
23.6
31.3
38.9
46.5
54.0
61.4
68.7
75.9
90.7
104.2
117.8
14
255
33*8
42.0
50.2
58.3
66.3
74.2
82.1
98.0
112.8
127.6
15
27.3
36.2
45*
53.8
62.6
71.2
79.7
85.3
88.2
105.4
121.3
137.4
i6
29.1
38.7
48.1
57.5
66.9
76.1
94.3
112.7
129.9
147.3
17
3'°
41. 1
51.2
61.2
71.1
81.0
90.8
100.5
120.0
138.5
157.1
i8
32.8
436
54.3
64.9
75.4
85.9
96.3
106.6
127.4
147. 1
166.9
19
346
46.0
57.3
68.6
797
90.8
101.8
112.8
1347
155.7
176.7
. 20
1
36.5
48.5
60.4
72.3
84.0
95.7
107.3
118.9
142.0
164.3
186.5
' 21
38.3
50.9
63.5
75.9
88.3
100.6
112.9
125.0
149.4
172.9
196.4
1 22
40.2
531
66.5
79.6
92.6
105.5
118.4
131.2
156.7
181. 5
206.2
23
42.0
55.8
69.6
83.3
96.9
"0.5
123.9
1373
164.0
190.1
^^5;0
i 24
43.8
f^§
72.7
87.0
101.2
115.4
129.4
1434
171.4
198.7
225:8
. 25
457
60.8
7SZ
90.7
105.5
109.8
120.3
135.0
149.6
178.7
207.2
235.6
26
47.5
63.2
78.8
943
125.2
140.5
'§52
186.1
215.8
245.4
27
49.4
^§•7
81.9
98.0
114.1
130. 1
146.0
161.8
193.4
224.4
2553
28
512
68.1
85.0
101.7
118.4
135.0
151.5
168.0
200.7
233.0
265.1
29
530
70.6
88.0
105.4
122.7
139.9
157.0
174.1
208.1
241.6
274.9
30
549
730
91. 1
109. 1
127.0
144.8
162.6
180.2
215.4
250.2
284.7
31
56.7
755
94.2
112.8
131.3
149.7
168.1
186.4
222.7
258.8
294.5
: 32
58.6
77.9
97.2
1 16.4
135.6
1546
173.6
192.5
230.1
267.4
304.3
33
60.4
80.4
100.3
1 20. 1
1399
159.5
179.1
198.7
204.8
237.5
276.0
314.2
34
62.2
82.8
103.4
123.8
144.2
164.5
184.7
244.8
284.6
324.0
35
64.1
IH
106.4
127.5
148.5
169.4
190.2
210.9
252.2
293.1
333.8
36
65.9
87.8
109.5
131.2
152.7
174.3
1957
217.1
259.5
301.7
343.6
38
69.6
92.7
115.6
138.5
161.3
184. 1
206.8
229.3
274.3
3189
363.2
40
733
97.6
121.8
145.9
169.9
193.9
217.8
241.6
289.0
336.1
382.9
42
77.0
102.5
127.9
153.3
178.5
203.7
228.8
253.9
3037
353.3
402.5
Jl
825
109.8
137. 1
164.3
191.2
218.5
245.4
272.3
325.8
379.1
432.0
88.0
117.2
146.3
175.4
203.8
233.2
262.0
290.7
347.9
404.8
461.4
5«
93.6
124.6
1555
186.4
216.5
247.9
278.6
309.1
370.0
430.6
490.9
54
99.1
131.9
164,7
197.5
229.2
262.6
295.1
327.5
392.1
456.4
520. J
57
104.6
139.3
173.9
208.5
241.8
277.4
311.7
345.9
414.2
482.1
549.8
60
IIOlI
146.6
183. 1
•
219.6
254.5
292.1
328.3
364.3
436.3
507.9 579.3
256
WEIGHT OF METALS.
Table No. 86 {continued),
Bv External Diameter, i Foot Long.
Ext.
1
Thicknrss in Inchbs.
DiAM.
3/i6
%
5/16
H
7/16
>i
9/16
H
^
Ji
z
ft. in.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
53
1.03
1.08
1.44
1.71
2.06
2.39
2.74
3.08
3.42
4.09
4.77
543
56
1.50
1.80
2.16
2.50
2.87
3.22
3.58
4.29
5.00
570
59
'•'2
1.55
1.88
2.26
2.62
300
337
3.75
449
523
5.96
60
1. 18
1.61
1.96
2.36
2.74
3 14
3.52
3.91
469
546
6.22
63
1.23
1.67
2.05
2.45
2.85
327
3.66
4.08
4.88
5.69
6.49
66
1.28
1.73
2.13
2.55
2.97
340
3.81
4.24
5.08
5.92
6.75
69
'•33
1.78
2.21
2.65
3.09
353
3.96
4.41
5.28
6.38
7.01
70
1.38
1.84
2.29
2.75
3.20
3.66
4.10
4.57
557
7.28
76
1.48
1.95
2.46
2.95
3.43
3.92
4.39
4.90
5.87
6.84
7.80
80
1.58
2.07
2.62
3.15
3.67
4.19
4.69
5.23
6.26
7.30
8.33
Thick
NBSS IN Inches.
^%
iX
1^8
^%
im:
2
{ cwts.
2X
2>4
2^
3
3>^
4
inches.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
cwts.
6
.481
.520
.557
.592
.652
.701
'W.
.761
6>i
•530
•575
.618
.657
.729
.789
.838
.872
.906
7
•579
.630
.678
.723
.805
.876
.938
.982
1.03
1.05
VA
.629
.68s
.738
.789
.882
.964
1
1.04
1.09
I.I5
1. 18
8
.678
.740
.799
.855
.959
i.os
1. 14
1.20
1.27
1.32
1.38
8>^
•727
.794
.859
•^11
1.04
1. 14
1.23
I.3I
1.39
'•45
1.53
9
.777
.849
.919
.986
I. II
1.23
133
1.42
I.5I
1.58
1.69
1.75
9y2
.826
.904
.980
1.05
I.I9
1 M'
1.43
1.53
1.63
I.7I
1.84
1.93
10
.875
.959
1.04
1. 12
1.27
1.40
1.53
1.64
1.75
1.84
1.99
2.10
II
.974
1.07
i.x6
X.25
1.42
1.58
1.73
1.86
199
2.10
2.30
2.46
12
1.07
I.18
1.28
1.38
1.57
1 1.75
1.92
2.08
2.23
2.37
2.61
2.81
13
1.17
1.29
1.40
1.51
173
1.93
2.12
2.30
2.47
2.63
2.92
3.16
H
1.27
1.40
1.52
1.64
1.88
2.10
2.32
2.52
2.71
2.89
3.22
l&
15
1.37
1.51
1.65
1.78
2.03
2.28
2.52
2.74
2.95
3.16
\^
16
1.47
1.62
1.77
I.9I
2.19
2.45
2.71
2.96
319
312
4.21
17
1.57
1.73
1.89
2.04
2.34
2.63
2.91
3.18
3.44
3.68
4.14
4.56
18
1.66
1.84
2.01
2.17
2.49
2.81
3."
3.40
3.68
395
4.45
4.91
20
1.86
2.06
2.25
2.43
2.80
3.16
350
3.83
4.16
4.47
5^
5.61
22
2.06
2.27
2.49
2.70
3."
l^
3.90
4.27
4.64
5.00
5.68
6.32
24
2.26
2.49
2.73
2.96
3.41
4.29
4.71
5.12
5.52
6.29
7.01
27
255
2.82
3.09
3.35
3.87
4.38
4.88
5.37
1^5
6.31
7.21
8.06
30
2.85
3.»S
346
3.75
433
4.91
547
6.03
6.57
7.10
8.13
9.12
33
3.14
3.48
3.82
4.14
4.79
544
6.06
6.68
7.29
7.89
9.05
10.2
36
344
3.81
4.18
4.54
5.25
5.96
6.66
7.34
8.01
8.68
9.97
II. 2
39
374
4.14
4.54
4.93
5.72
6.49
7.2s
8.00
8.74
9.47
10.9
12.3
42
403
4.47
4.90
533
6.18
7.01
7.84
8.66
9.46
10.3
II. 8
13.3
45
4.33
479
5.26
572
6.64
Z5*
8.43
9.31
10.2
II. I
12.7
14.4
48
4.62
5.12
5.62
6.12
7.10
8.07
9.02
9.98
10.9
II. 8
13.7
15.4
51
4.92
5^5
5.98
6.51
7.56
8.59
9.61
10.6
1 1.6
12.6
14.6
16.5
54
5.22
5.78
6.35
6.91
8.02
9.12
10.2
"3
J2.4
134
15.5
17.5
57
551
6.11
6.71
7.30
8.48
9.64
10.8
11.9
^K
14.2 i 16.4
18.6
60
5.81
6.44
7.07
7.70
8.94
10.2
11.4
12.6
13.8
15.0 173
19.6
CASTIRON CYLINDERS.
257
Table No. 86 {continued).
By External Diameter, i Foot Long.
Ext.
DiAM.
i%. in.
59
60
63
66
69
70
76
80
86
90
96
100
106
11 o
116
12 o
130
140
150
160
17 o
180
190
200
THICKNESS IN Inches.
^%
cwts.
6.10
6.40
6.70
7.00
7.29
7.58
7.88
8.17
8.77
9.36
995
10.5
II. I
11.7
12.3
12.9
135
14. 1
15.3
16.5
177
18.8
20.0
21.2
22.4
23.6
iX
cwts.
6.77
7.09
7.42
775
8.08
8.41
8.74
9.07
9.72
10.4
II. o
11.7
12.3
13.0
137
143
15.0
15.6
16.9
18.3
19.6
20.9
22.2
23.5
24.8
26.1
^H
CWIS.
7.43
779
8.15
8.51
8.88
9.24
9.60
9.96
10.7
1 1.4
12. 1
12.9
13.6
14.3
150
15.7
16.5
17.2
18.6
20.1
21.5
23.0
24.4
25.9
27.3
28.8
iK
cwts.
8.09
8.48
8.88
9.27
9.67
0.1
0.5
0.9
1.6
2.4
32
4.0
4.8
5.6
6.4
7.2
79
8.7
20.3
21.9
235
25.0
26.6
28.2
29.8
314
i^
cwts.
9.40
9.86
0.3
0.8
1.2
1.7
2.2
2.6
35
4.5
54
6.3
7.2
8.1
9.1
20.0
20.9
21.8
237
25.5
27.3
29.2
31.0
32.9
34.7
36.5
2
^%
cwts.
cwts.
10.7
12.0
II. 2
12.6
II. 8
13.2
".3
138
12.8
•14.4
13.3
14.9
13.9
15.5
;t^
16. 1
154
^*1W
16.5
J^S
17.5
19.7
18.6
20.8
19.6
22.0
20.7
23.2
21.7
24.4
22.8
2S.6
23.8
26.7
24.9
27.9
27.0
30.3
29.1
32.7
31.2
35.0
33.3
?H
35.4
;39.8
37.5
42.2
396
44.5
41.7
46.9
^}i
cwts.
133
13.9
14.6
15.2
159
16.6
17.2
17.9
21.5
23.1
24.4
25.7
27.1
28.4
29.7
31.0
336
36.3
38.9
41.5
44.2
46.8
494
52.0
»ii
cwts.
14.5
15.2
159
cwts.
15.8
16.6
cwts.
18.3
cwts.
17
2^^S
WEIGHT OF METALS.
Table No. 87. — ^Volume and Weight of CastIron Balls.
Given the Diameter.
Diameter.
Contents.
Weight.
Diameter.
Contents.
Weight.
Diameter.
Contents. Weight.
inches.
cubic
inches.
pounds.
inches.
cubic
inches.
pounds.
inches.
cubic feet.
cwts.
I
.524
.136
8
268.1
69.8
19
2.078
8.35
1/3
1.77
.460
sy2
321.5
•83.7
20
2.424
974
2
4.19
1.09
9
381.7
99.4
21
2.806
11.28
2y2
8.18
2.13
9}4
448.9
I16.9
22
3.227
12.97
3
3}^
14. 1
3.68
585
10
523.6
136.4
23
3,688
4.188
14.82
y
22.5
inches.
cubic feet
cwts.
24
16.83
4
33.5
8.73
II
•403
1.62
25
4736
19.03
4>^
477
12.4
12
.524
2.10
26
5327
21.40
5 ^
655
17.0
13
.666
2.68
27
5963
23.96
sH
87.1
22.7
14
.832
334
28
6.651
26.72
6
1131
295
15
1.023
4. 1 1
29
7.390
29.69
6}^
1438
37.5
16
1. 241
4.99
30
8.181
32.87
7
179.6
46.8
17
1.489
598
31
9.027
36.27
1/2
220.9
57.5
18
1.767
7.10
32
9930
3990
Nbie. — To find the weight of balls of other metals, multiply the weight given in the
table by the following multipliers : —
For Wrought Iron 1.067, making about 7 per cent. more.
Steel 1.088 ,, 9 „
Brass 1.12 ,, 12 ,,
Gun Metal 1. 165 „ 16)^ „
Table No. 88. — Diameter of CastIron Balls.
Given the Weight.
.Weight
Diameter.
Weight
Diameter.
Weight
Diameter.
Weight
Diameter.
pounds.
inches.
pounds.
inches.
•
pounds.
inches.
cwis.
inches.
>^
154
14
4.68
80
8.37
8
18.73
I
194
16
4.89
90
8.71
9
19.48
2
3
4
5
2.45
2.80
3.08
332
18
20
25
28
509
527
5.68
590
100
9.02
10
12
14
16
20.17
21.44
22.57
23.60
cwts.
I
inches.
937
10.72
6
3.53
30
6.04
2
11.80
18
24.54
7
372
40
6.64
3
1351
20
25.42
8
389
50
7.16
4
14.87
25
27.38
9
4.04
56
743
5
16.02
30
29.10
10
4.19
60
7.60
6
17.02
35
30.64
12
4.45
70
8.01
7
17.91
40
32.03
WEIGHT OF FLAT BAR STEEL.
259
Table No. 89. — Weight of Flat Bar Steel.
I Foot Long.
Width in Inches.
Thickness.
>i
H
•
H
I
iX
i>^
iH
mches.
lb.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
X
.425
.533
.640
.743
.850
1.06
1.28
1.49
s/i<
•53i
.665
.800
.929
1.06
1.33
159
1.86
^
.638
.798
.960
1. 1 1
1.28
"•59
1.91
2.23
7/16
.744
.931
I.I2
1.30
1.49
1.86
2.23
2.60
}i,
.850
1.06
• 1.28
1.49
1.70
2.13
^•55
2.98
9/16
1.20
1.44
1.67
I.91
2.39
2.87
335
><,
^_
133 •
1.60
1.86
2.12
2.66
3 19
372
T
1.76
2.04
2.34
2.92
35*
4.09
—
1:92
2.23
2.5s
3.19
3.83
4.46
t'
—
—
—
2.41
2.76
345
4.14
4.83
—
2.60
2.98
372
4.46
5^1
»5/i6
—
—
319
3.98
4.78
5.58
I
— —
—
—
340
4^25
5.10
595
Width ii
4 Inches.
Thickmkss.
Jb ^^ *^»4^ ** Awa^k^v
2
2X
2)i
2H
3
3X
3>i
4
inches.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
H,
1.70
I.9I
^•33
2.34
2.55
2.76
2.98
340
s/i6
2.13
2.39
2.87
2.66
2.92
3 19
3.45
372
4.25
H
2.98
3.19
3.51
3.83
4.14
4.46
5.10
1/16
3.35
372
4.09
4.46
4.83
5.21
^P
yi
3.40
383
4.25
4.68
5.10
553
595
6.80
9/16
3.83
430
4.78
5.26
5.74
6.22
6.69
7.6s
H
4.25
4.78
53'
5.84
6.38
6.91
7.44
8.50
T
4.68
5.26
5.84
6.43
7.01
7.60
8.18
9.35
5.10
5.74
6.38
7.01
7.65
8.20
8.93
10.2
«3/i«
5.53
6.22
6.91
7.60
8.29
8.98
9.67
II. I
n
5.95
6.69
744
8.18
8.93
9.67
10.4
II.9
r^'^ \ It
7.17
797
8.77
9.56
10.4
II. 2
12.8
7.65
8.50
9.35
10.2
II. I
11.9
136
^P** *>««^ ^v ^ tf <*
Width n
r Inches.
4>i
5
S}i
6
eyi
7
7K
8
iiicii<es. 1 Ite
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
5^ 1 3.82
4.26
4.68
§•'2
5.52
5.96
6.38
6.80
V.<s 4.78
§•32
5.84
6.38
t'^
744
7.97
8.50
H 5.74
6.38
7.02
7.66
8.92
8.28
8.92
9.56
10.2
7/16 6.70
7.44
8.18
9.66
10.4
II. 2
II.9
1 5^ 1 7'^
8.50
9.36
10.2
II. I
II. 9
12.8
136
9/z6 8.60
9.56
10.5
II.5
12.4
13.4
143
iS3
^ 9.56
10.6
n.7
12.8
13.8
14.9
159
17.0
i«/i6
10.5
11.7
12.9
14.0
^H
16.4
17.5
18.7
yr 1
II.5
12.8
14.0
153
16.6
17.9
19. 1
20.4
>3/i6
12.4
13.8
15.2
16.6
18.0
193
20.7
22.2
■*■/■ «^
^
134
14.9
16.4
17.9
19.4
20.8
22.3
23.8
«5/x6
14.3
15.9
'25
19. 1
20.8
22.4
23.9
25.6
I
15.3
17.0
18.7
20.4
22.1
23.8
25.5
27.2
26o
WEIGHT OF METALS.
Table No. 90. — ^Weight of Square Steel.
I Foot in Length.
Size.
inches.
V16
/A
V.6
H
V.6
'V.6
Weight.
pounds.
•053
.119
.212
•333
.478
.651
.850
1.08
1.33
1. 61
1.92
2.24
2.60
Size.
inches.
'A6
>^
v.6
I/'
/'4
H
v.6
2
Weight,
pounds.
3.06
340
3.83
4.30
4.79
5.31
5.86
6.43
703
7.65
8.30
8.98
979
Size.
inches.
2
2}i
2^
2^
2^
3
Weight.
pounds.
10.4
II. 2
II. 9
12.8
13.6
154
17.2
19.2
21.2
235
257
28.2
30.6
Size.
inches.
4
%
4K
4'
6
Weight.
pounds.
359
41.6
47.8
544
61.4
68.9
76.7
85.0
937
102.8
112.4
122.4
Table No. 91. — Weight of Round Steel.
I Foot in Length.
Diameter.
Weight.
Diameter.
Weight.
Diameter.
Weight.
Diameter.
Weight,
inches.
pounds.
inches.
pounds.
inches.
pounds.
inches.
cwts.
'A
.0417
I V.6
5.18
4
42.7
12
3.433
V.6
.0939
^H
6.01
^y^
48.3
I2>^
3729
H
.167
I V.6
6,52
A%
546
13
4.029
V.6
.260
Ifi
7^o5
4H
60.3
is'A
4.345
H
•375
I 'V.6
7.61
5
66.8
14
4.682
V.6
•511
I^
8.18
1%
73^6
I4j4
5013
y2
.667
I 'V.6
8.77
sy
80.8
IS
. 5364
v.6
.845
I^
9.38
SVa
88.3
^sH
5728
'V.6
1.04
I 'V.6
lO.O
6
96.1
• 16
i6>^
17
6.103
6.471
6.868
7.302
1.27
^•50
1.76
2
10.7
12.0
13.6
inches.
eyi
7
8
cwts.
1.007
I 168
'V.6
2.04
2.35
2^
16.7
I.34I
1.526
18
19
7724
8.607
I
2.67
2fi
18.4
sy
^•723
20
9537
I V16
3.00
2J^
20.2
9 ,
^•931
21
10.52
i}i
3^38
2^1
22.0
9y
2.152
22
1154
I VI6
3^76
24.1
10
2.385
23
12.62
I^
4.17
3^
28.3
loj^
2.629
24
1373
I V16
4.60
3>^
32.7
II
2.884
I?^
505
3?<
34^2
ii>^
3^i5o
WEIGHT OF CHISEL STEEL.
261
Table No. 92. — Weight of Chisel Steel — Hexagonal, Octagonal,
AND OvalFlat.
I Foot in Length.
Dxaneter
across the
Sides.
HbX AGONAL SbCTION,
Octagonal Section.
Sectional Area.
Weight.
Length
to weigh
I cwt.
Sectional Area.
Weight.
Leng^th
to weigh
I cwt
inches.
square inches.
pounds.
feet
square inches.
pounds.
feet
H
.1217
.2165
.3383
.4871
.6631
.414
.736
I.I5
1.66
2.25
245
138
88.3
61.3
45
.1164
.2070
.3236
.4659
.6342
.396
.704
I.IO
1.58
2.16
268
151
96.5
67
49.3
''H
. .8661
1.096
2.94
373
34.5
27.3
.8284
1.048
2.82
3.56
377
30
1.353
1.637
1.949
4.60
557
6.63
22.5
18.3
153
1.294
1.566
1 1.864
4.40
5.32
6.34
24
20
16.8
t
OvalFlat Section.
inch. inch.
I x)^
.2510
.4463
.6974
.853
1.52
2.37
119
67
43
Table No. 93. — Weight of one Square Foot of Sheet Copper.
To WireGauge employed by Williams, Foster, & Co.
Specific Weight taken as 1. 16 (Specific Weight of Wrought Iron = i).
Thickness.
Weight of!
X Square i
Foot
Thickness.
Weight of
I Square
Foot.
Thickness.
Weight of
X Square
Foot
Wire
Gauge.
No.
Inch
(approxi
mate).
pounds.
Wire
Gauge.
No.
Inch
(approxi
mate).
pounds.
Wire
Gauge.
No.
Inch ^
(approxi
mate).
pounds.
I
.306
14.0
II
.123
565
21
.0338
155
2
.284
13.0
12
.109
5.00
22
.0295
135
3
.262
12.0
13
.0983
4.50
23
.0251
iiS
4
.240
II.
14
.0882
4.00
24
.0218
I.OO
5
.222
10.15
15
.0764
3.50
25
.0194
.89
6
7
.203
.186
930
8.50
16
17
.0655
.0568
3.00
2.60
26
27
.0172
.0153
•79
.70
8
.168
7.70
18
.0491
2.25
28
.0135
.62
9
•153
7.00
19
.0437
2.00
29
.0122
•56
10
.138
6.30
20
.0382
1.75
30
.0110
•50
i WEIGHT OF METALS,
Table No. 94. — Weight of Copper Pipes and Cylinders,
BY Internal Diameter.
Length, i Foot. Thickness by HoIuapHera WiteGauge (Table No. 13).
Spedfic Weighti.l6 (Specific Wdght of WroughtIroni).
Di^H.
Ito.
Ibt
ibi.
Ila.
Ibi.
Ita.
lbs.
Lbs.
lbs.
ibt.
H
3 "4
2.84
233
1.92
1.53
I.4I
1.21
1.05
■934
f?i
%
384
349
2.91
2.44
1,99
1.84
1.60
1.41
1.27
H
454
4.13
349
295
245
2.27
1.77
1.60
143
'A
5^3
4.78
4.06
347
2.91
2.71
239
2.13
•93
173
H
593
5.42
6.07
4.64
399
337
383
3 '4
2.7E
It
2.26
2.04
6.63
522
4.50
357
3. "7
2.60
2.3s
7.32
6.71
579
S.02
429
4.00
3.57
3.22
2.93
2.66
'
7.36
6.37
553
4.74
4.43
396
357
326
2.97
iH
8.71
8.00
6.95
6.05
S.20
4. 86
435
394
3.60
3.28
"X
9.4D
8,6s
7.52
6.57
ii\
S29
475
430
m
]:g
'H
9.30
8.10
7.08
S72
514
4.66
t'A
lo's
994
8.68
7,60
6.57
6,16
553
S02
4.60
4.20
'H
ii.S
10.6
9.26
8.12
7.02
659
593
539
493
til
iH
9.83
8.63
7.48
7.02
6.32
575
S27
m
ia!s
11.9
10,4
915
7.93
;:si
6.71
6.11
5.60
S.I2
'3.5
12.S
9.66
839
7.11
6.47
593
543
2H
14.2
i3.a
11.6
10.2
8.84
8.31
6.83
6.27
574
i%
14.9
138
10.7
93°
8.74
7 "9
6.60
6.05
2H
'56
145
12,7
975
9 '7
756
6.94
6.36
'H
16.3
IS 1
13.3
niy
9.60
7.92
7.27
6.67
'H
17.0
158
•39
122
10.7
10.0
9.07
8. 38
7.60
',%
^H
17.7
16.4
•45
ii^S
10.5
947
8.64
7.94
aji
.§.4
17. 1
iSo
133
138
IIS7
10.9
9.86
9.00
8.27
7 59
3
"91
177
15.5
"■3
10.3
9.36
8.61
7.90
3J4
iO.4
19.0
16.8
148
12.9
ii,2
U.I
10.1
9.17
8.52
3H
21.8
179
•59
51
131
11.8
10.8
994
913
3K
232
21.6
19. 1
16.9
m
12.6
iis
ia.6
975
4
24.6
22.9
17.9
'57
•34
12.3
"3
10.4
4X
15.9
24.2
21.4
19,0
16.6
:«
14Z
130
12.0
110
A'A
It',
r,
22.5
'^5
'52
'37
12.7
11.6
^M
237
248
18.4
17.4
IS.8
14.4
133
5
301
28,0
22.1
19.3
18.Z
16.6
ISI
14.0
12.8
%%,
32.8
293
26.0
231
10.2
19. 1
173
159
14.6
'35
s'A
30.6
27.1
24.1
18. t
16.6
«S3
141
SU
34.2
319
28.3
25.2
2a!8
.8.9
;i:5
16.0
14.7
6
356
332
29.5
26.2
23.0
'■•'
197
16.6
'53
S.04,
532
WEIGHT OF COPPER PIPES AND CYLINDERS.
263
Table No 94 {contintied).
Length, i Foot. Thickness by Holtzapffers Wire Gauge (Table No. 13).
Specific Weight = 1. 16 (Specific Weight of Wrought Iron= i).
Thick
kiss.
8
9
10
II
la
13
14
15
16
17
18
19
ao
W.G.
Inch.
.165
.148
•134
.120
.109
.095
.083
.072
.065
.058
.049
.042
•035
"/64*
9/6</
lbs.
9/64^
H^
7/64
3/3a/
lbs.
5/64^
lbs.
'A6/
Vi6^
3/64/
3/64^
^M
IST.
DlAM.
,bches.
Ihs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
.581
.491
.422
•357
.310
.254
.210
.173
.150
.129
.104
.086
.068
X
.832
.716
.626
.540
.476
.398
.336
.282
.249
.217
.178
.149
.121
H
1.08
.941
.830
.722
.641
•553
.462
•391
.348
•305
.253
.213
!228
a
133
I.I7
1.03
.904
.807
.687
.588
.500
.447
.393
.327
•277
H
1.58
1.39
1.24
1.09
.972
.831
•Z'4
.610
.545
.481
.402
.341
.281
)i.
183
1.62
1.44
1.27
1. 14
.975
.840
.719
.644
.570
.476
.404
•334
}i
2.09
1.84
1.65
1.45
1.30
1. 12
.966
.828
•743
.658
•550
.468
•387
I
2.34
2.05
1.85
1.63
1.47
1.26
1.09
.938
.842
.746
.625
.532
.440
iH
^:i^
2.27
2.05
1.82
1.63
1. 41
1.22
1.05
.940
.834
.699
.596
•493
iH
2.49
2.26
2.00
1.80
155
1.34
1. 16
1.04
.922
•774
.659
•547
'^,
309
2.72
2.46
2,18
1.97
1.70
1.47
1.27
1. 14
I.OI
.848
.723
.600
'K
3.34
2.94
2.67
2,36
2.13
1.84
1.60
1.38
1.24
1. 10
.922
•787
.653
•K
359
317
2.87
2.55
2.30
1.99
1.72
1.48
134
1. 19
•997
.851
.706
%
3.&4
339
307
2.73
2.46
2.13
1.85
1.59
1.43
1.27
1.07
•9^5
•759
^a
4.09
3.62
3.28
2.91
2.63
2.27
197
1.70
^53
1.36
1.15
.978
.812
2
434
3.84
3.48
309
2.79
2.42
2.10
1. 81
1.63
1.45
1.22
1.04
.865
t%
459
407
3.69
327
2.96
2.56
2.23
1.92
H^
1.54
1.29
i.ii
.919
i'4
484
429
3.89
345
3 12
2.71 ;
^•35
2.03
1.83
1.63
1.38
1.17
.972
%
509
4.52
4.09
364
329
2.85
2.48
2.14
193
H'
145
123
1.03
1.08
^}i
534 4.74
4.30
3.82
345
3.00
2.60
2.25
2.03
1.80
1.53
1.30
»H
ls
497
4.50
4.00
362
3.14
2.73
2.36
2.13
1.89
1.60
1.36
1. 13
^H
519
4.71
418
379
3.28
2.86
2.47
2.22
1.98
1.68
143
1. 18
m
6.09
5.42
4.91
4.37
395
343
2.98
2.58
2.32
2.07
H^
1.49
1.24
3
634
5.66
5.11
455
4.12
3.57
3"
2.69
1
2.42
2.16
1.82
155
1.29
^
6.8s
6.11
5.52
4.91
445
3.86
3.36
2.91
2.62
2.33
1.96
1.68
1.40
3^
7.8s
6.56
593
5.28
4.78
4.15
3.61
3.12
2.82
^•5J
2. 1 1
1.81
1.51
3¥
7.01
6.33
5.64
5.11
4.44
3.87
3.34
3.01
2.68
2.26
1.94
1.62
4
835
7.46
6.74
O.OI
5.44
4.73
4.12
3.56
321
2.86
2.41
2.06
173
4)(
8.85
79»
7.14
6.37
577
5.02
4.37
3.78
341
3<H
2.56
2.19
1.84
*^/
935
8.36
8.81
755
6.74
6.10
530
4.62
400
3.61
3.21
HI
2.32
1.94
aH
9.85
7.96
7.10
6.43
559
487
4.22
3.80
339
2.86
2.45
2.05
5
10.4
9.26
8.36
7.46
6.77
5.88
5.^3
4.44
4.00
356
301
2.57
2.16
5^
10.9
9.71
8.77
7.83
7.10
6.17
5.38
4.66
420
374
3. IS
2.70
2.27
5^
11.4
ia2
9.18
8.19
7.43
6.46
I'M
4.88
4.40
392
3.30
2.83
2.38
I^
11.9
10.6
9.58
8.56
7.76
6.7s
5.09
4.59
4.09
3.45
2.96
2.48
6
•M
II. I
999
8.92
8.09
7.04
6.14
5.31
4.79
4.27
36o
309
2.58
262
WEIGHT OF METALS.
Table No. 94. — ^Weight of Copper Pipes and Cylinders,
BY Internal Diameter.
Length, i Foot. Thickness by Holtzapffel's WireGauge (Table No. 13).
Specific Weight = 1. 16 (Specific Weight of WroughtIron= i).
Thick
KESS.
W. G.
Inch.
Int.
DlAM.
inches.
H
H
n
1%
i}i
iH
2
2)i
2H
2H
2>^
2H
2H
2H
3
3X
3'A
3H
4
A'A
aH
5
5^
SA
sH
6
0000
•454
39/64
lbs.
3H
384
454
523
593
6.63
7.32
8.02
8.71
9.40
10. 1
10.8
11.5
12. 1
12.8
>3.S
14.2
14.9
15.6
16.3
17.0
17.7
18.4
19. 1
20.4
21.8
23.2
24.6
25.9
273
28.7
30.1
3'S
32.8
34.2
356
000
00
.425
'7/64/
.380
lbs.
lbs.
2.84
349
4.78
2.33
2.91
349
4.06
5.42
6.07
6.71
7.36
4.64
5.22
5.79
6.37
8.00
8.65
9.30
9.94
6.95
752
8.10
8.68
10.6
II. 2
11.9
12.S
9.26
9.83
10.4
II.O
13.8
H5
15.1
1 1.6
12. 1
12.7
13.3
15.8
16.4
17.1
17.7
13.9
'4.5
15.0
156
19.0
20.3
21.6
22.9
16.8
17.9
19. 1
20.2
24.2
25.4
26.7
28.0
21.4
22.5
23.7
24.8
29.3
30.6
319
332
26.0
27.1
28.3
29.5
.340
11/32
lbs.
1.92
2.44
2.95
347
399
4.50
5.02
553
6.05
6.57
7.08
7.60
8.12
8.63
9.15
9.66
10.2
10.7
II. 2
11.7
12.2
12.8
13.3
138
14.8
15.9
16.9
17.9
19.0
20.0
21.0
22.1
23.1
24.1
25.2
26.2
.300
'9/64/
lbs.
1.53
1.99
2.45
2.91
3.37
3.83
4.29
4.74
5.20
5.65
6. 1 1
6.57
7.02
7.48
7.93
8.39
8.84
930
975
10.2
10.7
II. I
11.57
12.0
12.9
139
14.8
15.7
16.6
18.4
19.3
20.2
21. 1
22.1
23.0
.284
9/32/
lbs.
1. 41
1.84
2.27
2.71
3.14
357
4.00
443
4.86
5.29
572
6.16
6.59
7.02
74S
7.8g
8.31
8.74
9.17
9.60
lO.O
10.5
10.9
"•3
12.2
131
'32
14.8
15.6
16.5
17.4
18.2
19. 1
20.0
20.8
21.7
.259
X/
lbs.
1. 21
1.60
2.00
2.39
2.78
317
357
396
435
4.75
5.14
553
593
6.32
6.71
7.II
7.50
7.89
8.29
8.68
9.07
9.47
9.86
10.3
1 1. 1
11.8
12.6
»34
14.2
15.0
15.8
16.6
17.3
18. 1
18.9
19.7
.238
'5/64/
lbs.
1.05
I.4I
1.77
2.13
2.50
2.86
322
3.57
394
430
4.66
5.02
5.39
5.75
6. 1 1
6.47
6.83
7.19
756
7.92
8.28
8.64
9.00
936
10. 1
10.8
II. 5
12.3
13.0
137
14,4
15. 1
159
16.6
'Z3
18.0
.220
7/3"/
lbs.
.934
1.27
1.60
1.93
2.26
2.60
2.93
326
3.60
393
4.26
4.60
4.93
5.27
5.60
593
6.27
6.60
6.94
7.27
7.60
7.94
8.27
8.61
9.27
994
10.6
"•3
12.0
12.7
13.3
14.0
14.6
153
16.0
16.6
.203
'3/64
lbs.
.809
1. 12
L43
1.73
2.04
2.35
2.66
2.97
3.28
3.58
3.89
4.20
4Si
4.82
5.12
543
5.74
6.05
6.36
6.67
6.97
7.28
7.59
7.90
8.52
9 13
9.75
10.4
II.O
II. 6
12.2
12.8
13.5
14. 1
14.7
15.3
.180
3/16 b.
lbs.
.667
.941
1. 21
1.49
1.76
2.03
2.31
2.58
2.85
313
340
3.68
3.95
4.22
4.50
477
5.04
532
5.86
6.14
6.41
6.68
6.95
7.50
8.04
8.59
9.»3
9.67
10.2
10.8
"3
11.9
12.4
12.9
135
WEIGHT OF COPPER PIPES AND CYLINDERS.
263
Table No 94 {continual).
Length, i Foot. Thickness by Holtzapffers Wire Gauge (Table No. 13).
Specific Weight = 1. 16 (Specific Weight of Wrought Iron= i).
Thick
KBSS.
W. G.
•
8
9
10
IZ
12
13
14
15
.072
5/64^
lbs.
z6
.065
Vi6y
17
z8
Z9
20
Inch.
.165
lbs.
.148
9/64/
lbs.
.134
9/64^
.120
.109
7/64
.095
3/3>/
1
lbs.
.083
lbs.
.058
.049
3/64/
.042
3/64^
.035
Int.
DiAM.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
inches.
i
.581
.832
1.08
1.33
.491
.716
.941
I.17
.422
.626
.830
1.03
•357
.540
.722
.904
.310
.476
.641
.807
•254
.398
'.687
.210
.336
.462
.588
•173
.282
•391
.500
.150
.249
.348
.447
.129
.217
•305
•393
.104
.178
.253
.327
.086
.149
.213
.277
.068
.121
•'75
.228
i
I
1.58
183
2.09
2.34
139
1.62
1.84
2.05
1.24
1.44
1.65
1.85
1.09
1.27
1.63
.972
1. 14
1.30
1.47
.831
.975
1. 12
1.26
.714
.840
.966
1.09
.610
.719
.828
.938
•545
.644
.743
.842
.481
.570
.658
.746
.402
.476
.550
.625
•341
.404
.468
.532
.281
.334
.387
.440
2.59
2.84
309
334
2.27
2.49
2.72
2.94
2.05
2.26
2.46
2.67
1.82
2.00
2.18
2.36
1.80
1.97
2.13
I.4I
1.55
1.70
1.84
1.22
134
1.47
1.60
1.05
1. 16
1.27
1.38
.940
1.04
1.14
1.24
•834
.922
I.OI
1. 10
.699
.774
.848
.922
.596
.659
.723
.787
•493
.547
.600
.653
2
4.09
434
317
339
362
3.84
2.87
307
328
3.48
2.55
2.73
2.91
309
2.30
2.46
2.63
2.79
1.99
2.13
2.27
2.42
1.72
1.85
1.97
2.10
1.48
1.59
1.70
I.81
134
1.43
1.53
1.63
1. 19
1.27
1.36
1.45
•997
1.07
i.iS
1.22
.851
.978
1.04
.706
.759
.812
.865
2H
459
4.84
5.09
5.34
4.07
4.29
474
369
3.89
4.09
4.30
327
345
364
3.82
2.96
3.12
329
3.45
2.56
2.71
2.85
3.00
2.23
2.35
2.48
2.60
1.92
2.03
2.14
2.25
'•Z3
1.83
193
2.03
154
1.63
1.29
1.38
1.45
153
I. II
1.17
123
1.30
.919
.972
1.03
1.08
3
6.09
6.34
4.97
5.19
5.42
5.66
4.50
4.71
4.91
5.II
4.00
4.18
4.37
4.55
362
379
3.95
4.12
3 '4
3.28
343
357
2.73
2.86
2.98
3."
2.36
2.47
2.58
2.69
1
2.13
2.22
2.32
2.42
2.07
2.16
1.60
1.68
1.82
1.36
143
1.49
1.55
1. 18
1.24
1.29
3U
3%
3H
4
6.85
7.85
8.35
6.1 1
6.56
7.01
7.46
5.52
5.93
6.33
6.74
4.91
5.28
5.64
O.OI
4.45
4.78
5.11
5.44
3.86
415
444
4.73
3.36
36i
3.87
4.12
2.91
312
334
356
2.62
2.82
3.01
3.21
2.33
2.51
2.68
2.86
1.96
2. 1 1
2.26
2.41
1.68
1. 81
1.94
2.06
1.40
1.51
1.62
1.73
4X
4K
4H
5
8.85
935
9.85
10.4
7.91
8.36
8.81
9.26
7.14
7.55
7.96
8.36
6.37
6.74
7.10
7.46
6.10
6.43
6.77
5.02
530
55§
5.88
4.37
4.62
487
5.13
3.78
4.00
4.22
4.44
341
3.61
3.80
4.00
3.04
3.21
3.39
356
2.56
2.71
2.86
3.01
2.19
2.32
2.45
2.57
1.84
1.94
2.05
2.16
10.9
11.4
11.9
12.4
9.71
ia2
10.6
II. I
8.77
9.18
9.58
9.99
7.83
8.19
8.56
8.92
7.10
7.43
7.76
8.09
6.17
6.46
6.75
7.04
5.38
6.14
466
4.88
5.09
531
4.20
4.40
459
4.79
3^74
392
4.09
427
3.15
330
3.60
2.70
2.83
2.96
309
2.27
2.38
2.48
2.58
264
WEIGHT OF METALS.
Table No. 94 {continued).
Length, i Foot. Thickness by Holtzapffel's WireGauge (Table No. 13).
Specific Weight=i.i6 (Specific Weight of Wrought Iron =1).
Thick
ness.
0000
000
00
z
2
3
4
5
6
7
W. G.
•454
»6/64
.425
.380
.340
.300
.284
.259
.238
.220
.203
.180
Inch.
'7/64/
^/
"/33
19/64/
9/3^/
x/
'5/64/
Vs"/
^3/64
3/16 b.
Int.
•
DiAM.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
inches.
eVz
38.4
35.8
31.8
28.3
21.3
19.5
18.0
14.6
7
41. 1
38.3
34.1
30.3
26.6
25.1
22.8
20.9
19.3
17.8
157
VA
439
40.9
36.4
32.4
28.4
22.4
20.6
16.8
8
46.6
435
38.7
34.5
303
28.6
26.0
23.8
22.0
20.2
179
9
52.1
48.7
433
38.6
33.9
32.0
29.1
26.7
24.6
22.7
20.1
10
57.7
53.8
479
^•2
37.5
355
32.3
29.6
273
25.2
22.2
11
^2^
590
52.5
46.8
41.2
38.9
354
32.5
30.0
27.7
24.4
12
68.7
64.2
57.2
51.0
44.8
42.4
38.6
35.4
32.7
30.1
26.6
13
74.2
69.3
61.8
55.1
48.5
45.8
41.7
38.3
35.3
32.6
28.8
14
79.7
74.5
66.4
592
52.1
49.3
44.9
41.2
38.0
351
31.0
15
85.2
P't
71.0
63.4
55.8
52.7
48.0
44.1
40.7
37.6
332
16
90.7
84.8
75.6
67.7
59^4
56.2
51.2
46.9
434
40.0
354
17
96^3
90.0
80.2
71.8
63.0
59.6
54.3
49.8
46.0
42.5
37.5
18
101.8
95 1
84.9
76.0
66.7
63.1
57.4
52.7
48.7
45.0
39.7
19
107.3
100.3
89.5
80.1
70.3
66.5
60.6
5g.6
51.4
47.4
41.9
20
112.8
105.5
94.1
84.2
74.0
70.0
63.7
58.5
54.0
499
44.1
21
118.3
1 10. 7
98.7
88.3
77.6
73.4
66.9
61.4
56.7
52.4
46.3
22
123.8
115.8
103.3
92.5
81.3
76.9
70.0
64.3
59.4
54.9
48.5
23
129.3
120.9
107.9
96.6
84.9
80.3
732
67.2
62.1
57.3
50.7
24
I34^8
1 26. 1
112.6
100.6
88.6
83.8
76.3
70.1
64.7
59.8
52.9
26
146.0
136.4
121. 8
108.8
95.9
90.7
82.6
759
70.1
64.7
57.2
28
^IZ*
146.7
1310
117.1
103. 1
97.6
89.0
81.7
Z54
69.7
61.6
30
168.4
157.1
140.2
125.4
1 10.4
104.5
95.3
87.5
80.8
74.6
66.0
32
179.6
167.4
149.5
133.6
117.7
111.4
101.6
93.3
86.2
79.6
70.4
34
190.7
^IV
158.7
141.9
125.0
1 18.3
107.9
99.1
91.5
84.5
747
36
201.9
188.0
167.9
150. 1
132.3
125.2
114.2
104.9
96.9
89.5
79.1
WEIGHT OF COPPER PIPES AND CYLINDERS.
265
Table No. 94 {continued).
Length, i Foot. Thickness by Holtzapffers WireGauge (Table No. 13).
Specific Weight = 1. 16 (Specific Weight of Wrought Iron= i).
Thick
mess.
8
9
10
zz
Z2
13
14
15
16
17
18
19
20
W. G.
Inch.
.165
.148
.134
.120
.109
.095
.083
.072
.065
.058
.049
.042
•035
"/fi*^
9/64/
9/64^
7/64
3/3>y
sM
5/64^
V.6/
V16*
3/64/
3/64*
V3«/
Int.
DiAM.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
inches.
e>%
13.4
12.0
10.8
9.65
8.75
7.61
6.64
5.75
5 '2
462
390
334
2.80
7
14.4
12.9
II. 6
10.4
9.42
8.19
7.14
6.19
5.58
497
420
3.60
301
2^
15.4
13.8
12.47
II. I
10. 1
8.77
7.65
6.63
5.98
5.33
449
3.85
323
8
16.4
147
132
II. 8
10.74
9.34
8.15
7.06
6.37
5.68
479
4.10
3.43
9
18.4
'f5
14.9
^33
12. 1
10.5
9.16
7.94
7.16
6.38
5.39
4.61
3.86
10
20.4
18.2
16.5
148
13.4
II. 7
10.2
8.81
7.95
7.08
5.98
5.12
4.28
II
22.4
20.0
18. 1
16.2
147
12.8
II. 2
9.69
8.74
779
6.58
5.63
470
12
24.4
21.8
19.8
17.7
16.0
140
12.2
10.6
9.53
8.49
7.18
6.14
5.13
13
26.4
23.6
21.4
19. 1
'Z4
15.1
132
11.4
10.3
9.20
7.77
6.65
5.55
14
28.4 1 25.4
23.0
20.6
18.7
16.3
14.2
12.3
II. I
9.90
In
7.16
5.98
15
30.4
27.2
24.6
22.1
20.0
17.4
15.2
132
II.9
10.6
8.96
7.67
6.40
16
32.4
29.0
26.3
23.5
21.3
18.6
16.2
141
12.7
"3
9.56
8..18
6.82
17
344
30.8
27.9
25.0
22.7
19.7
17.2
149
13.5
12. 1
10.2
8.69
7.27
18
36.4
32.6
29.5
26.4
24.0
20.9 1 18.2
1'
15.8
14.3
12.7
10.7
9.20
7.69
19
384
34.4
31.2
27.9
25.3
22.0
19.2
16.7
I5.I
134
II3
9.71
8.12
20
40.4
36.2
32.8
293
26.6
23.2
20.2
17.6
159
14. 1
11.9
10.2
8.54
21
42.4
38.0
34.4
30.8
27.9
24.3
21.3
18.4
16.6
14.8
12.S
10.7
8.96
22
4*.4
39.8
36.0
32.3
293
25.5
22.3
19.3
17.4
155
131
II. 2
9.39
23
46.4
41.6
37.7
33.7
30.6
26.7
233
20.2
18.2
16.2
137
11.8
9.81
24
48.5
434
39.3
35.2
319
27.8
24.3
21. 1
19.0
16.9
143
12.3
10.2
26
52.5
47.0
42.6
38.1
34.6
30.1
26.3
22.8
20.6
18.4
15.5
133
II. I
28
56.5
50.6
45.8
41.0
37.2
32.4
28.3
24.6
22.2
19.8
16.7
143
11.9
30
60.5
54.2
49.1
43.9
39.9
347
30.3
26.3
237
21.2
»7.9
15.3
12.8
32
^^
57.8
52.3
46.8
42.5
37.0
32.3
28.1
25.3
22.6
19. 1
16.3
13.6
34
68.5
61.4
55.6
49.8
^51
39.4
344
29.8
26.9
240
20.3
17.4
145
36
1
1
1
72.5
65.0
58.8
52.7
47.8
41.7
36.4
31.6
28.5
25.4
21.5
18.4
15.3
266
WEIGHT OF METALS.
Table No. 95. — ^Weight of Brass Tubes,
BY External Diameter.
Length, i Foot. Thickness by Holtzapffel's WireGauge (Table No. 13).
Specific Weight=i.ii (Specific Weight of Wrought Iron=i).
Thick
NBSS.
W. G.
15
z6
17
18
19
20
ax
aa
23
24
25
vInch.
.072
5/64^
.065
.058
'/16 b.
.049
3/64/
.042
3/64 b.
.035
V32/
.032
' V32
.028
1/32 b.
.025
1.6/64
.022
1.4/64
.020
1.3/64
DiAM.
inches.
lbs.
lbs.
lb.
lb.
lb.
lb.
lb.
lb.
lb.
lb.
lb.
3/16
s/16
.201
.187
.087
.130
.172
.079
.115
.150
.072
.102
.132
.037
.062
.088
."3
.o3
!o8i
.104
.031
.052
.072
.092
.029
.047
.065
.083
.026
.042
.058
.074
.024
•039
.053
.068
7/16
9/16
.306
.358
.411
.234
.281
.329
•376
.214
.256
.298
.340
.186
.221
.257
.293
.163
•»93
.224
.254
.138
.164
.189
•215
.128
.151
.174
.197
."3
•133
•154
.174
.102
.120
.138
.156
.090
.106
.122
.138
.082
.097
.III
.126
^3/16
•463
.515
.567
.620
.423
.470
.517
.564
.382
•424
.467
.509
.328
.364
•399
.435
.285
.346
.376
.240
.265
.291
.316
.221
•244
.267
.290
.194
.215
•235
•255
.174
.192
.211
.229
.154
.170
.186
.202
.141
.155
.170
.184
»5/i6
I
.672
.724
.'S8i
.611
.658
.706
.801
•S5I
.593
.635
.719
.471
.506
.542
.613
.407
.437
.468
.529
.342
•367
•393
.443
.314
•337
.360
.407
.276
.296
.316
•357
.247
.265
. .283
.320
.218
.234
.250
.282
.199
.213
.228
•257
.986
1.09
1.20
.896
.991
1.09
.972
.684
.827
.590
.651
.712
.494
.545
.596
.453
.546
.398
.439
.479
.356
.392
•429
•314
.346
.378
.286
•315
.344
W. G.
9
ID
XX
12
13
14
15
16
17
18
19
Inch.
.148
9/64/
lbs.
.134
9/64^.
.120
yib.
.109
7/64
.095
3/3a/
.083
s/64/
.072
5/64 b.
.065
.058
x/x6 b.
.049
3/64/
.042
3/64*.
DiAM.
inches.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
1.90
2.II
2.33
2.54
1.74
1.93
2.13
2.32
'•5f
1.76
1.94
2.12
1.45
1.60
1.76
1.92
1.28
1. 41
1.55
1.69
113
1.25
137
1.49
.986
.991
1.20
1.30
.896
.991
1.09
1. 18
.804
.888
•972
1.06
.684
•755
.827
.898
.590
.651
.712
.773
2
2>^
a. 76
2.97
319
340
2.52
2.71
2.91
3.10
2.30
2.47
2.65
2.83
2.08
2.24
2.39
2.55
1.83
1.97
2.10
2.24
1. 61
1.85
1.97
1.40
1. 61
1.72
1.28
^•37
1.56
1.14
1.23
131
139
.969
1.04
i.ii
1.18
.834
•895
.956
1.02
2X
2^
2^
3.62
3.83
4.04
3.30
3.49
3.69
3.01
319
3.37
2.71
2.86
302
2.38
2.52
2.66
2.09
2.21
2.33
1.82
1.93
2.03
1.66
1.85
1.48
1.56
1.65
1.25
133
1.40
1.08
1. 14
1.20
WEIGHT OF BRASS TUBES.
267
Table No. 95 (continued).
Length, i Foot. Thickness by Holtzapffel's WireGauge (Table No. 13).
Specific Weight = I. H (Specific Weight of Wrought Iron=i).
Thick
NBSS.
W. G.
3
4
5
6
7
8
9
ID
IX
Z2
13
.259
.238
.220
.203
.180
.165
.148
.134
.120
.109
•095
Inch.
x/
'5/64/
7/3>/
'3/64
3/16 ^.
"/64 *•
9/64/
9/64^.
>^^.
7/64
3/32
DiAM.
inches.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
lbs.
2
5.24
4.87
^55
4.24
3.80
3.52
3.19
2.91
2.65
2.39
2.10
2yi
5.62
522
4.87
454
4.07
376
3.40
3«o
2.83
2.55
2.24
2H
599
557
519
4.83
433
4.00
3.62
330
3.01
2.71
2.38
2H
6.37
591
551
5.13
459
4.24
3.83
349
3 19
2.86
2.52
^H
6.75
6.26
583
5.42
4.85
4.48
4.04
369
337
3 02
2.66
^^
7.12
6.60
6.14
5.72
5">
4.72
4.26
3.88
355
3.18
2.79
2^
7.50
6.95
6.47
6.01
5.38
4.96
4.47
4.07
373
334
2.93
^H
7.88
730
6.79
6.31
5.64
5.20
4.69
4.27
391
3.50
307
3
8.25
7.64
7.II
6.60
590
S44
4.90
4.46
4.09
3.66
3.21
3V
9.01
8.33
7.75
7.19
6.43
5.92
5.49
4.85
4.43
3.98
3.48
3>^
9.76
9.02
8.39
7.78
6.95
6.40
6.07
524
4.78
430
3.76
3H
10.5
9.72
9.03
8.37
747
6.88
6.65
'563
5.12
4.61
4.04
4
"•3
10.4
9.67
8.96
8.00
7.36
7.24
6.02
5.46
4.93
4.31
4^
12.0
ii.i
10.3
955
8.52
7.83
7.82
6.41
5.80
5.25
4.59
^H
12.8
11.8
10.9
10. 1
9.04
8.31
8.41
6.80
6.15
5§S
4.87
aH
13.5
12.5
II.6
10.7
9.56
8.79
8.99
7.19
H^
5.88
514
5
14.3
132
12.2
"3
10. 1
9.27
957
7.58
6.83
6.20
5.42
5^
^^•?.
X39
12.9
11.9
10.6
9.75
10.2
7.97
7.17
6.51
5.69
^'4
15.8
14.6
13.5
12.5
II.I
10.2
10.7
8.36
7.52
6.83
597
sH
16.5
153
14. 1
131
II. 7
10.7
"3
8.75
7.86
7.1S
6.25
6
17.3
159
14.8
13.7
12.2
II. 2
11.9
9.14
8.20
7.46
6.52
268
WEIGHT OF METALS.
Table No. 96. — Weight of One Square Foot of Sheet Brass.
Thickness by Holteapffeirs WireGauge (Table No. 13).
Weight of
Weight of
Weight of
Thickness.
I Square
Thickness.
X Square
Thickness.
T Square
Foot.
Foot.
Foot.
No.W.G.
inch.
pounds.
N0.W.G.
inch.
pounds.
N0.W.G.
■inch.
pounds.
3
•259
10.9
II
.120
505
19
.042
1.77
4
.238
lO.O
12
.109
4.59
20
•035
1.47
5
.220
9.26
13
•095
4.00
21
.032
135
6
.203
8.55
14
.083
349
22
.028
i.i8
7
.180
7.58
15
.072
3.03
23
.025
105
8
.165
6.95
16
.065
2.74
24
.022
.926
9
.148
6.23
17
.058
2.44
25
.020
.842
10
.134
SM
18
.049
2.06
Table 97. — Size and Weight of Tin Plates.
Mark.
Size of Sheets.
Number
of Sheets
in a Box.
Weight
per Sox.
IC
IX
IXX
IXXX
IXXXX
inches, inches.
14 X 10
» >l
if >»
it If
sheets.
225
>»
99
pounds.
112
140
161
182
203
SDC
SDX
SDXX
SDXXX
S D XXXX
15 X II
>f 9f
» if
» »
200
168
189
210
231
252
DC
DX
DXX
DXXX
DXXXX
17 X 12}^
100
98
126
147
168
169
WEIGHT OF TIN AND LEAD PIPES.
269
Table No. 98. — ^Weight of Tin Pipes,
As manufactured.
I FOOT IN LENGTH.
Diameter
Externally.
inches.
1/
n
H
1 1
■i
2
Thickness.
3/32" inch.
lbs.
.148
.384
.620
.856
1.095
1.328
1.564
1.802
^ inch.
lbs.
.472
.787
T.IO3
I.417
1732
2.047
2.362
Diameter
Externally.
inches.
2l'
2^
3
Thicknbss.
}i inch.
lbs.
504
567
6.30
6.93
7.56
8.19
Table No. 99.— Weight of Lead Pipes.
As manufactured.
!
Bore.
Length.
Weight and Thickness of Pipe.
Calcu
Calcu
Calcu
Calcu
Weight.
lated
Thick
ness.
1 Weight
lated
Thick
ness.
Weight.
lated
Thick
ness.
Weight
lated
Thick
ness.
inches.
feet
lbs.
inch.
lbs.
inch.
lbs.
bch.
lbs.
inch.
>^
IS
14
.097
16
.112
18
.124
22
.146
^
n
17
.101
21
.121
30
.140
Va
w
24
.112
28
.147
32
.181
36
•215
I
»
36
.136
1 42
.156.
56
.200
64
.225
^Va
12
36
.139
42
.160
48
.180
52
•193
^%
»
48
.156
56
.179
72
.224
84
.257
^H
w
72
.199
84
.228
96
.256
2
»
72
.178
84
1
.204
96
.231
112
.266
2^
10
i 84
.200
96
.227
112
.261
3
»
112
.218
140
•275
3J^
»
130
.225
160
•273
4
»
170
.257
220
•327
4j^
19
170
.232
220
•295
4H,
4'A
V.6ii
ich thick. W<
sight pel
9J
r lineal f
bot
.... 22.04. lbs.
99
.... 23.25 ff
4H
5
99
99
.... 2*1. (
56 „
99
TV jy
270
WEIGHT OF METALS.
Table No. loo. — Dimensions and Weight of Sheet Zinc.
( VielieMontagne.)
SiZBS OP
Shebts.
Weight
No.
Thickn«»«
9.0 X. 50
metres;
3.0 X. 65
metres;
9. ox. 80 metres;
per
area, x square metre.
area, 1.3 sq., metres.
6. 56 X 2. X3 feet; area.
area, 1.6 sq. metres.
6 '56X9. 62 feet; area.
square
toot.
6. 56 X X. 64 feet ; area,
X0.76 square feet.
13.99 square feet.
X7. 29 square feet.
No.
inxn.
inch.
kUs.
lbs.
kUs.
lbs.
kils.
lbs.
lbs.
9
.41
.0161
2.90
6.39
3.70
8.16
4.6
10.14
.589
lO
•51
.0201
345
7.61
4.45
9.81
5.5
12.12
.704
II
.60
.0236
4.05
8.93
5.30
11.68
6.5
14.33
.832
12
.69
.0272
4.65
10.25
6.10
13.45
7.5
16.53
.960
13
.78
.0307
530
11.68
6.90
15.21
85
18.74
1.088
14
.87
•0343
5.95
1312
7.70
^5*§^
95
20.94
I.216
^5
.96
.0378
6.55
14.44
8.55
18.85
10.5
23.15
1.344
i6
1. 10
.0433
7.50
16.53
975
21.50
12.0
26.46
1.536
17
1.23
.0485
8.45
18.63
10.95
24.14
13.5
29.97
1.740
i8
1.36
.0536
935
20.61
12.20
26.90
15.0
33.07
1.920
19
1.48
.0583
10.30
22.71
13.40
29.54
16.5
36.38
2.II2
20
1.66
.0654
11.25
24.80
14.60
32.19
18.0
39.68
2.304
21
1.85
.0729
12.50
27.56
16.25
3582
20.0
44.09
2.560
22
2.02
.0795
13.75
30.31
17.90
39.46
22.0
48.50
2.816
23
2.19
.0862
15.00
3307
19.50
42.99
24.0
52.91
3.073
24
2.37
•0933
16.25
35.82
21.10
46.52
26.0
57.32
3.329
^5
2.52
.0992
17.50
38.58
22.75
50.15
28.0
61.73
3.585
26
266
.1047
18.80
41.44
24.40
53.79
31.0
68.34
3.969
Table No. loo {continued).
Special Sizes for Sheathing Ships.
Sizes op
Sheets.
Weight
No.
Thickness.
X.1SX.35 metres;
1.30 X. 40 metres;
per
area, .402 sq. metre.
] 3.77 X X. 15 feet; area,
433 sq feet.
area, .520 sq. metre.
4. 26X1. 31 feet; area,
5.60 sq. feet.
square
foot.
No.
mils.
inch.
i kUs.
lbs.
kils.
lbs.
lbs.
15
.96
.0378
2.65
5.84
3.40
W
1.344
16
1. 10
.0433
3.00
6.61
3.90
8.60
1.536
17
1.23
.0485
3.40
7.50
4.40
9.70
1.740
18
1.36
■0536
3.75
8.27
4.90
10.80
1.920
19
1.48
.0583
, 4.15
9.15
5.35
11.79
2.II2
20
1.66
.0654
; ^55
10.03
5.85
12.90
i 2.304
1
Note. — A deviation of 25 dekagrammes, or about halfapound, more or less, from the
proper weight of each number of sheet, is allowed.
Nos. I to 9 are employed for perforated articles, as sieves, and for articles de Paris,
Nos. 10 to 12 are used in the manufacture of lamps, lanterns, and tinware generally, and
for stamped ornaments. The last numbers are used for lining reservoirs, and for baths
and pumps.
FUNDAMENTAL MECHANICAL PRINCIPLES.
FORCES IN EQUILIBRIUM.
Solid Bodies.
Paralldogram of Forces. — ^When a body remains at rest whilst being acted
on by two or more forces, it is said to be in a state of equilibrium, and so also
are the forces. Thus, if the forces p/, q ^, r r, Fig. 86, acting on the body
pqr^ keep it at rest, they are in
equilibrium, and any two of them
balance the third. The lines of
force, if produced, meet at one point
o within the body, and if a parallel r
(^ram be constructed having two
adjacent sides proportional to and
parallel to two of the forces respec Fig. 86.— EquiUbrium of Forces.
tively, to represent them in magni
tude and direction, the diagonal of the parallelogram will represent the third
force in magnitude and direction. Let the lines o p, OQ, Fig. 87, represent
the forces p/, q^ in magni
tude and direction, and com :p
plete the parallelogram • by
drawing the parallels p r, q r, Bf_„
and draw o r. Then o r re
presents in magnitude and
direction the resultant of the ^
two forces; and RO taken in Fig. 87.— Parallelogram of Forces.
the opposite direction repre
sents the third force Rr, Fig. 86. If it be applied in this direction to
the point o, as indicated by a dot line o r', it would balance the other two.
This construction is called the paralldogram of forces y and is applicable to
any three forces in equilibrium.
Three forces in equilibrium may also be represented by a triangle, or
half a parallelogram. For example, the triangle o p r represents by its
three sides the forces o P, o q, o r, the side p r being substituted for o q.
Three forces in equilibrium must be in the same plane.
When the directions of three forces holding a body at rest, and also the
magnitude of one of them, are known, the magnitudes of the other two can
be determined by constructing a parallelogram in the manner above exem
plified, and measuring the lengths of the 5ide^ apd the diagonal.
2/2
FUNDAMENTAL MECHANICAL PRINCIPLES.
Polygon of Forces, — Equilibrium may subsist between more than three
forces, which need not necessarily be in the same plane; and they can, like
those already illustrated, be developed in direction and magnitude by diagram.
Thus, let the point o, Fig. 88, representing a solid body, be kept at rest by a
number of forces, op, o q, o r, o s, o t. Find the equivalent diagonal o/ for
the first two forces ; then construct the parallelogram and diagonal o r for the
resultant of op and the third force or; and again the parallelogram and
diagonal o s for the resultant of o r and the fourth force o s. The last
resultant o^ represents in one the four distributed forces op, oq, or, os, and
it balances the fifth force o t equal and opposite to it. A5 o j and o t are
in the same straight line, their resultant is of course nothing.
The several forces thus dealt with may be combined into a polygon of
forces. Draw o p. Fig. 89, parallel and equal to o p, Fig. 88, p q parallel and
equal to o Q, Q r parallel to o r, r s
parallel to os; then, finally, so,
completing the polygon, will be
parallel and equal to ot. Fig. 88,
the last of the series. Professor
Mosely illustrates the polygon of
forces by the united action of a
number of bellringers, pulling by a
number of ropes joined to a single
rope. The polygon constructed as
in Fig. 90, shows successively by
corresponding letters, the individual
contributions of the bellringers,
combined into one vertical force.
Again, equilibrium may be estab
lished between a number of forces
Fig. 88.— Equilibrium of more than Three Forces.
acting in the same plane, applied to different points in a body, or system of
bodies. For example, let the forces p o, q o, r o, s o, and t o, be applied
to several points, o, o, o, o, o, on a flat board
ABC, Fig. 91, by means of cords and weights;
it will settle into a position of equilibrium,
when the opposing forces arrive at a balance
between themselves. An axis or pivot may
be established at any point, m, on the surface
of the board, without disturbing the equilib
rium, and it may be viewed as a centre of
motion round which the forces tend to turn
the board, some in one direction, the others
the opposite way, balancing each other. The
effect of each force to turn the body about
the centre is represented by the product of its magnitude by the leverage^
or perpendicular distance of its direction from the centre; draw these
perpendiculars, and multiply each force by its perpendicular or leverage,
then the resulting products will be divisible into two sets, tending to turn
the board in opposite directions. The sum of the first set of products is
equal to the sum of the second set, as is proved by the fact of equilibrium.
Moments of Forces, — ^The product of a force by the perpendicular dis
tance of its direction from any given point, is called the momait of the
Fig. 89. — Polygon of Forces.
FORCES IN EQUILIBRIUM.
273
force about that point; and the equilibrium above discussed, in connection
with Fig, 91, is the result of the equality of moments.
The law of the equality of moments may be thus set forth; — If any
number of forces acting anywhere in the
same plane, on the same body or connected
system of bodies, be in equilibrium, then
the sum of the forces tending' to turn the
system in one direction about any point in
that plane, is equal to the sum of the mo
ments of those forces tending to turn the
system in the other direction.
Such balanced forces may be transferred
to a single point, and placed about it, as in
l~ig. 88, parallel to their directions as they
stand; and they will continue in equilibrium,
holding the point at rest. A polygon of
the iotctspgrsl within Fig. 90, may be
constructed similarly to Fig. 89.
Though the principle of the polygon of
forces be sufficient to test the equilibrium
of a system of forces acting at one point,
yet the principle of the equality of moments, Fig. 90— Beunngen, Polygon of Forcei.
in addition, is necessary to establish the
equilibrium of a system applied to different points. The two principles
conjointly are necessary, and they are sufficient, as conditions of equilibrium.
The Catenary. — When
a chain, or a rope, or a
flexible series of rods, is
suspended by its extremi
ties, supporting weights
distributed along its
length, in a state of rest,
ii forms a polygon of
forces in equilibrium, as
in Fig. 92. If all the
forces except those which o
act on the extremities of
the chain, be combined
into a resultant, then the
two extreme sides being
produced, will meet the
direction of the resultant
at one point Thus, in
the polygon, Fig. 92,
loaded with weights, w,w,
&C., the verrical resultant Fig. 91.— Equjiiry of Moments.
of these weights w' W,
passing through their common centre of gravity, intersects at w' the two
extreme sections p a, p' b, when produced downwards.
Similarly, in the catenary, Fig. 93, which is the curve assumed by a rope
or other flexible medium uniformly loaded and suspended by the two
274
FUNDAMENTAL MECHANICAL PRINCIPLES.
extremities, if tangents be drawn to the extremities a, b, of the curve, meeting
at w*, they represent the directions of the forces sustaining the curve at
Fig. 9a.— The Catenary.
Fig. 93. — ^The Catenary.
those points, and they intersect at the same point w*, the vertical line G w^
passing through tlie centre of gravity of the curve. Let the weight of the
•^
flf
T
Fig. 94.— Centrifugal Forces in Equilibrium.
Fig. 95.— Parallel Forces in Equilibrium.
curve be represented by g w^, and complete the parallelogram m n, then
w* M and w*' n represent in force and direction the tension at the points
B and A.
Centrifugal Forces in Equilibrium. — ^\Vhen a cylindrical vessel is exposed
to a uniform internal pressure, as the pressure of steam within a boiler, for
example, the pressure is balanced by the resistance of cohesion of the
material of the boiler. Let a b c D, Fig. 94, be the section of a cylindrical
boiler, the radial pressure of the steam may be represented by the arrows,
which are equal and opposite in direction. The tension on the metal in
resisting the internal pressure at any particular section b i>, is equal to the
sum of the pressures resolved into the direction at right angles to b d, or
parallel to ac, according to the triangles, or halfparallelograms of force
attached to each oblique arrow. The total vertical pressure thus obtained
by the resolution of forces is equal to the total vertical pressure which
FORCES IN e;quilibrium.
27s
would be exerted on the sectional line B d if it be supposed to be a rigid
diaphragm across the boiler, which is easily calculated. If the radial
pressure be, for example, 100 lbs. on each square inch of surface, then the
total pressure, or the tension on the two sides at b and d, would be
100 X BD on each inch of length of the two sides; that is to say, if the
diameter b d be equal to 60 inches, the tension on the two sides would be
60 X 100=6000 pounds for each inch of length.
A similar argument applies to the tension on the rim of a revolving fly
wheel.
Parallel Farces. — Systems of parallel forces are particular cases of the
forgoing. — Let a, b, c, d, e, f, Fig. 95, be a system of parallel forces in equili
brium; and MN a line perpendicular to them in the same plane, and cut
by them at the points a^byCyd^e,/, They may act at any points in their
lines of direction without disturbing the equilibrium, and they may be sup
posed to be applied at those points in the line m n. Then, the sum of the
moments of the three forces a, b, c, acting in one direction, with respect to
any point m as a centre, is equal to the sum of
the moments of the forces d, e, f, opposed to
them. Further, the sum of the simple forces
A, B, c, irrespective of their moments, is equal to
that of the forces d, e, f, so that the fact of their
being in equilibrium resolves itself into a case of
action and reaction, for the two equivalent forces
representing the two opposing sums, act in the
same straight line in opposite directions.
When three parallel forces balance each other,
acting on a straight line, two of them must be
opposed to the third; and the third must act
between the other two, being equal and opposite
to their resultant. Let a, b, c, Fig. 96, be three
such forces applied to the line e g f, at the points
E, G, f respectively; then, with respect to the
point G, the moment of the force b is nothing, because it passes through
that point and has no leverage on it There remain the moments of the
extreme forces, a and c, which are equal to each other, that Is to say
axeg = cxfg.
Fig. 96.— Three Parallel Forces in
Equilibriuia.
From this it follows, by proportion, that
A : c : : FG : eg,
and that the extreme forces are to each other inversely as their distances
from the middle force.
In general, of three parallel forces acting in equilibrium on an inflexible
line, the first in order is to the third as the distance of the third from the
second, is to that of the first from the second.
The sum of the first and third is equal to the second; and when the
distances or leverages are equal, the first and third forces are equal to each
other.
If the position of the line e f be inclined to the direction of the three
forces, and changed to e' f'. Fig. 96, the forces A, b, c, continue in equilibrium;
2'j(>
FUNDAMENTAL MECHANICAL PRINCIPLES.
Fig. 97. — Parallelopiped of Forces.
for the perpendicular lines g e and g f continue, as before, to be the lever
ages of the extreme forces a and c, on the central point g.
When three forces not in the same plane act on one point, there cannot
be equilibrium between thenL Two of these may be reduced to their
resultant by parallelogram, and
this resultant reduced with the
third force to a final resultant
For example, let the lines op,
OQ, OR, Fig. 97, represent in
magnitude and direction three
forces not in one plane acting
on the point o. By parallel
ogram, o s is the resultant of the
two forces o p, o Q, and o t is the
final resultant of o s and the third
force o R. That is to say, o t is the resultant of the three given forces.
If parallelograms be formed from each two of the three forces, they form,
when duplicated, a parallelopiped of forces, of which the diagonal is found
by the final resultant o t, and the principle of the parallelopiped of forces
may be thus defined: — If three forces be represented in magnitude and
direction by three adjacent edges of a parallelopiped, their resultant is
represented in magnitude and direction by the adjacent diagonal of the
solid.
There must be at least four forces to produce equilibrium about a point,
when the forces are not in the same plane.
The triangle ost. Fig. 97, comprises in its three sides the resultant of
the first and second forces, the third force, and the resultant of the three.
If the first resultant o s be replaced by the two lines o Q and Q s, which
represent the first and second forces, they form the foursided figure o Q s t,
the polygon of the four equilibrating forces.
A greater number of forces than four acting on a point may be reduced
in like manner.
Fluid Bodies.
The characteristic property of fluids is the capability of transmitting the
pressure which is exerted upon a part of the surface of the fluid, in all
directions, and of the same intensity: — the same pressure per square inch or
per square foot.
The pressure of water in a vessel, caused by its own gravity, increases in
proportion to the depth below the surface; and the pressure on a horizontal
surface, say, the bottom, is equivalent to the weight of the superincumbent
column of water, and the intensity of the pressure is independent of the
form of the vessel. The same rule applies when the pressure is from below
upwards.
The same rule also applies when the surface is either vertical or inclined,
and the mean height of the superincumbent column of water is measured
by the depth of the centre of gravity of the given surface below the siurface
of the water.
The water in open tubes communicating with each other, when in a state
of equilibrium, stands at the same level in the tubes, whatever may be the
relative diameters of the tubes.
MOTION. — GRAVITY. 277
The height of the superincumbent column of water is called the head of
water.
The buoyancy^ or the upward force with which water presses a body
immersed in it, from below upwards, is equal to the weight of water dis
placed by the body, or of a quantity of water equal in volume to the sub
merged body, or submerged portion of a body. The resultant of the
upward pressure passes through the centre of gravity of the water displaced;
and also, when the floating body is at rest, through the centre of gravity of
th« body.
This resultant line is called the axis of floatation, and the horizontal
section of the body at the surface of the water is \ht plane of floatation.
Bodies float either in an upright position or in an oblique position. A
body floats with stability , when it strives to maintain the position of equili
brium, and when it can only be moved out of this position by force, and
will return to it when the force is withdrawn. The metacentre is the point
at which the axis of floatation intersects the axis of a symmetrical body, as a
ship, when inclined. If the metacentre lies above the centre of gravity of
the ship, the ship floats with stability; if below, the ship is unstable; if the
centres coincide, which they must do in a cylinder or a sphere, for example,
the body floats indifferently in any position.
For the weight, volume, and pressure of water and air, see Water and
Air as standards of measure^ page 124.
MOTION.
The motion of a body is uniform, when the body passes through equal
spaces in equal intervals of time.
Velocity is the measure of motion, and is expressed by the number of
feet or oflier unit of length moved through per second or other unit of time.
Motion is accelerated, when the body moves through continually increased
spaces in equal intervals of time, like a railway train starting from a station.
Motion is retarded, when the body moves through continually decreased
spaces in equal intervals of time, like a railway train arriving at a station.
The acceleration and retardation are uniform, when the spaces moved through
increase or decrease by equal successive amounts, like a body falling by
the action of gravity, or, on the contrary, projected upwards in opposition
to gravity.
GRAVITY.
When bodies fall freely near the surface of the earth, the motion, as
already said, is uniformly accelerated; equal additions of velocity being made
to the motion of the body in equal intervals of time.
During the ist second of time, the body, starting from a state of rest,
falls through 16.095 ^^^t, or, say 16.1 feet; during the 2d second, it
falls through three times 16.1 feet; during the 3d second, it falls through
five times 16.1 feet, and so oa The body having, in the ist second, fallen
through 16. 1 feet, from a state of rest, with a motion uniformly accelerated,
it would move through 32.2 feet in the next second, with the velocity
thus acquired, without any additional stimulus from gravity; that is to
say, the velocity acquired at the end of the ist second is 32.2 feet per second.
During the 2d second, it in fact acquires an additional velocity of 32.2 feet
per second, making up, at the end of this second, a final velocity of 64.4
2/8 FUNDAMENTAL MECHANICAL PRINCIPLES.
feet per second. In like manner the body acquires an additional velocity
of 32.2 feet per second during the 3d second, making a final velocity of
three times 32.2 feet, or 96.6 feet per second. And so on.
Each of these additional velocities is acquired in falling through 16.1 feet
additional to the space fallen through in virtue of the velocity acquired at
the beginning of each second.
The relations of height fallen^ velocity acquired^ and time of fally are
simply exhibited in the following manner : —
During the successive seconds the heights fallen through are consecutiv?ly
as follow: —
time, I, I, I, I second,
height of fall, 16. i, 16. i x 3, 16. i x 5, 16.1 x 7 feet
And reckoning the totals from the commencement of the fali,
total times, i, 2, 3, 4 seconds,
total height of fall^ 16.1, 16.1 x 4, 16. i x 9, 16. i x 16 feet.
or t6.i, 16. 1 X 2^, 16. 1 X 3^, 16. i x 4^ feet.
or 16.1, 64.4, i44'9> 257.6 feet
Showing that the total height fallen is as the square of the total time.
Again, during the successive seconds, the successive additional velocities
acquired are : —
time, I, I, I, I second,
velocities acquired, 32.2, 32.2, 32.2, 32.2 feet per second
And the total or final velocities acquired, reckoning from the commence
ment of the fall, are : —
total times, i, 2, 3, 4 seconds,
final velocities, 32.2, 32.2 x 2, 32.2 x 3, 32.2 x 4 feet per second.
or 32.2, 64.4, 96.6, 128.8 feet per second.
m
Showing that the velocity acquired is in direct proportion to the time of the
fall.
The above relations of time, height, and velocity are brought together
for comparison, thus : —
time as, i,
velocity acquired as, i,
height of fall as, i,
or as I,
Showing that, whilst the velocity increases simply with the time, the lieight
of fall increases as the square of the time, and as the square of the velocity.
The force of gravity is expressed by the velocity communicated by gravity
to a body falling freely in a second, namely, 32.2 feet per second, and is
symbolized by g.
The foregoing relations of time^ velocity^ and height of fall, are comprised
in the six following propositions with their answers — applicable to the
condition of a body falling freely. They are much used in mechanical
calculations.
I and 2. Given the time^ to find the velocity and the height,
3 and 4. Given the velocity ^ to find the time and the height,
5 and 6. Given the height, to find the time and the velocity.
2,
Zy
4, &C.
2
3»
4, &C
4,
9»
16, ^C.
2^
6 y
42, &c.
GRAVITY. 279
Rules for the Action of Gravity.
Putting /=the time of falling in seconds, v = the velocity in feet per
second, ^=the height of fall in feet, and ^= gravity or 32.2; then
Rule i. Given the time of fall, to find the velocity acquired by a falling
body. Multiply the time in seconds by 32.2, the product is the final
velocity in feet per second. Or
Z/ = 32.2 / ( I )
Rule 2. Given the time of fall, to find the height of the fall. Multiply
the square of the time in seconds by 16.1. The product is the height of
fall in feet. Or
^=16.1 /2 (2)
Rule 3. Given the velocity^ to find the time of falling. Divide the
velocity in feet per second by 32.2. The quotient is the time in seconds.
Or
/=^ (3)
32.2
Rule 4. Given the velocity^ to find the height of fall "due" to the
velocity. Square the velocity in feet per second, and divide by 64.4. The
quotient is the height of fall in feet Or
/i=g^ (4)
64.4
Rule 5. Given the height of fall, to find the time 6f falling. Divide the
height in feet by 16.1, and find the square root of the quotient The root
is the time in seconds. Or
/
or /
= K\/ h (s)
Rule 6. Given the height of fall, to find the velocity due to the height
Multiply the height in feet by 64.4, and find the square root of the product
The root is the velocity in feet per second.
Or, multiply the square root of the height in feet by 8.025; the product
is the velocity in feet per second.
Note. — It is usual to take the integer 8 only for the multiplier.
Sjonbolically, these operations are expressed as follows : —
z/ = 32.2\/^ — = a/64.4^ = 8.025 a/ ^
j^ 32.2 ^ ^
OT in a round number z^= 8 a/ h '. (6)
The above rules are applicable, inversely, to the motion of bodies pro^
jected upwards and uniformly retarded by gravity. The height to which a
body projected vertically upwards by an initial impulse, will ascend, is equal
to tie height through which the body must fall in order to acquire the
initial velocity, and the same rule (4) applies in these two cases.
28o
FUNDAMENTAL MECHANICAL PRINCIPLES.
The following table, No. loi, gives the velocity acquired by a falling body
in falling freely through a given height Table No. 102 gives, conversely,
the height of fall due to a given velocity. Table No. 103 gives the fall and
the final velocity due to a given time of falling freely.
Table No. loi. — ^Velocity acquired by Falling Bodies, due to Given
Heights of Fall.
r^= 8.025 >y/ ^
Hdghtof
Van
Vdodty
in Feet
per Second.
Height of
FalL
Velocity
in Feet
per Second.
Height of
FalL
Velocity
in Feet
per Second.
Hdgfatof
Vdodty
in Feet
per Second.
feet.
ft. per sec.
feet
ft. per sec
feet.
ft. per sec
feeL
ft. per sec
.01
.803
30
13.90
23
38.49
50
56.74
.02
1. 14
35
15.01
24
39.31
100
80.25
.03
139
4.0
16.05
25
40.12
150
98.28
.04
1.61
4.5
1703
26
40.92
200
"35
.05
1.80
50
17.99
27
41.70
300
139.0
.06
1.97
55
18.82
28
42.47
400
160.5
.07
2.12
6.0
19.66
29
43.22
500
179.9
.08
2.27
6.5
20.46
30
43.95
600
196.6
.09
2.41
7.0
21.23
31
44.68
700
212.3
.1
2.54
7.5
21.97
32
45.39
800
226.9
.2
3.20
8.0
22.69
33
46.10
900
240.7
•3
4.40
8.5
23.40
34
46.79
1000
2538
.4
5'07
9.0
24.07
35
47.47
1500
310.8
•5
5.68
95
2473
36
48.15
2000
3589
.6
6.22
10
25.38
37
48.81
2500
401.2
.7
6.71
II
26.62
38
49.47
3000
4395
.8
7.18
12
27.80
39
50.11
3500
4747
•9
7.61
13
28.93
40
50.75
4000
507.S
I.O
8.03
14
3003
41
51.38
4500
538.3
1.2
8.79
15
31.08
42
52.01
5000
5674
L4
9.50
16
32.10
43
52.62
6000
621.6
1.6
10.15
17
33.09
44
53.23
7000
671.4
1.8
10.77
18
34.05
45
53.83
8000
717.8
2.0
"35
19
34.98
46
5443
9000
761.3
2.25
12.04
20
35.89
47
55.02
1 0000
802.5
2.50
12.69
21
36.77
48
55.60
2.75
1331
22
37.64
49
56.17
GRAVITY.
281
Table No. 102. — Height of Fall due to Given Velocities.
h^
64.4"
Velocity
in Feet
Height of
Fall.
Velocity
in Feet
Height of
Fall
Velocity
in Feet
Height of
Fall
Velocity
in Feet
Height of
Fall.
per Second.
per Second.
per Second.
per Second.
fc per sec
feet.
(t per sec.
feet
ft per sec
feet
ft. per sec
feet
•25
.0010
19
5.61
46
32.9
73
82.7
•50
.0039
20
6.21
47
34.3
74
85.0
■75
.0087
21
6.85
48
35.8
75
87.4
1. 00
.016
22
7.52
49
37.3
80
99.4
125
.024
23
8.21
50
38.8
85
II2.2
150
.035
24
8.94
51
40.4
90
125.8
175
.048
25
971
52
42.0
95
140. 1
2
.062
26
10.5
53
436
JOG
1553
25
.097
27
II3
54
453
105
171. 2
3
.140
28
II. 2
55
47.0
no
187.9
35
.190
29
131 •
56
48.7
115
205.4
4
.248
30
14.0
57
504
120
223.6
45
.314
31
14.9
58
52.2
130
262.4
5
.388
32
159
59
54.1
140
304.3
6
.559
33
16.9
60
559
150
3494
7
.761
34
17.9
61
57.8
175
4755
8
.994
35
19.0
62
59.7
200
621
9
1.26
36
20.1
63
61.6
300
1397
JO
1.55
37
21.3
64
63.6
400
2484
XI
1.88
38
22.4
65
65.6
500
3882
J2
2.24
39
23.6
66
67.6
600
5590
13
2.62
40
24.9
67
69.7
700
7609
J4
304
41
26.1
68
71.8
800
9938
15
349
42
27.4
69
739
900
12578
16
398
43
28.7
70
76.1
1000
15528
17
4.49
44
30.1
71
78.3
'* 1
5'"^^
45
314
72
80.5
282
FUNDAMENTAL MECHANICAL PRINCIPLES.
Table No. 103. — Height of Fall and Velocity acquired, for
Given Time of Fall.
^ = 16.1 A' V^2,2t
'
Velocity
Velocity
Velocity
Time of
Height of
acquired in
Feet per
Time of
Height of
acquired in
Feet per
Time of
Height of
acquired in
Feet per
Fall
Fall.
Fall
FalL
Fall.
Fall.
Second.
Second.
Second.
seconds.
feet.
ft. per sec.
seconds.
feet.
ft. per sec.
seconds.
feet.
ft. per sec
I
16. 1
32.2
12
2318
386.4
23
8517
740.6
2
64.4
64.4
13
2721
418.6
24
9273
772.8
3
144.9
96.6
14
3156
450.8
25
10062
805.0
4
257.6
128.8
15
3623
483.0
26
10884
837.2
5
402.5
161.
16
4122
5152
27
II737
869.4
6
5796
193.2
17
4653
547.4
28
12622
901.6
7
788.9
225.4
18
5217
5796
29
13540
9338
8
1030
257.6
19
5812
611.8
30
14490
966.0
9
1304
289.8
20
6440
644.0
31
15473
998.2
10
1610
322.0
21
7100
676.2
32
16487
1030
II
1948
354.2
22
7792
708.4
ACCELERATED AND RETARDED MOTION IN GENERAL.
The same rules and formulas that have been applied to the action of
gravity are applicable to the action of any other uniformly accelerating
force on a body, the numerical constants being adapted to the force. If an
accelerating or retarding force be greater or less than gravity; that is to say,
than the weight of the body, the constants 16.1, 32.2, and 64.4 are to be
varied in the same proportion.
To do this, multiply the constant by the accelerating force, and divide
the product by the weight of the body. Let / be the accelerating force,
and w the weight of the body, then the constant becomes
16.1/ ^^ 32.2/ ^^ 64.4/.
or
or
w
w
w
(a)
and substituting this in the formulas (i) to (6) for gravity, the following
general rules and formulas are arrived at for the action of uniformly accel
erating or retarding forces. The rules are written for accelerating forces,
but they apply by simple inversion to retarding forces also.
General Rules for Accelerating Forces.
The accelerating force and the weight of the body are expressed in the
same unit of weight; and the space in feet, the time in seconds, and the
velocity in feet per second.
In the following rules the time during which a body is acted on by an
accelerating force is called tAe titne; the velocity acquired at the end of the
ACCELERATED AND RETARDED MOTION. 283
time is called the final velocity; the space traversed by the body during the
time is called the space; the accelerating force is called the force,
t  the time.
V = the final velocity.
s = the space.
/ = the force.
w = the weight.
Rule 7. Given the weig/it^ the force^ and the time; to find the final
velocity. Multiply the force by the time and by 32.2, and divide by the
weight The quotient is the final velocity. Or
Rule 8. Given the weighty the forccy and the time; to find the space.
Multiply the force by the square of the time and by 16. i, and divide by the
weight Or
. = 1^:11^ (8)
Rule 9. Given the weight, the filial velocity, and the force; to find the
time. Multiply the final velocity by the weight, and divide by the force,
and by 32.2. The quotient is the time. Or
. W V / X
32.2/
Rule 10. Given the weight, \ht final velocity, and the force; to find the
space. Miiltiply the weight by the square of the velocity, and divide by
the force, and by 64.4. The quotient is the space. Or
W 7/^ / V
'=e^f ('°>
Rule ii. Given the weight, the force, and the space; to find the time.
Multiply the weight by the space, and divide by the force; find the square
root of the quotient, and divide it by 4. The last quotient is the time in
seconds. Or
^Hyf^ (")
Rule i 2. Given the weight, the force, and the space; to find the final
velocity. Multiply the space by the force, and divide by the weight; find
the square root of the quotient, and multiply by 8. The product is the final
velocity. Or
'7^
V=%J^
(12)
w '
Rule 13. Given the weight, the space, and \he final velocity; to find the
force. Multiply the weight by the square of the final velocity, and divide
by the space, and by 64.4. The quotient is the force. Or
^=6iT* <'3)
284 FUNDAMENTAL MECHANICAL PRINCIPLES.
Rule 14. Given the weigJit^ timcy and final velocity; to find the force.
Multiply the weight by the square of the velocity, and divide by the space,
and by 32.2. Or
/=^, (X4)
32.2 /
Note I. When the accelerating or retarding force bears a simple ratio to
the weight of the body, the ratio may, for greater readiness in calculation,
be substituted in the quantities (a) representing the modified constants, for
the force and the weight. Suppose the accelerating force is a tenth part of
the weight, then the ratio is i to 10, and
16. 1 ,
__ = 1. 61,
10
^2.2
^ =3.22,
10
644 _
10
6.44;
and these quotients may be substituted for 16. i, 32.2, and 64.4 respectively
in the formulas for the action of gravity (i) to (6), to fit them for direct
use in dealing with an accelerating force onetenth of gravity, the height h
in those formulas, of course, being taken to mean space traversed.
Note 2. The tables, Nos. 101103, pages 280282, for the relations of
the velocity and height of falling bodies, may be employed in solving
questions of accelerating force generally.
Example, A ball weighing 10 lbs. is projected with an initial velocity of
60 feet per second on a level bowlinggreen, and the frictional resistance
to its motion over the green is i lb. . What distance will it traverse before
it comes to a state of rest? By rule 10,
10 lbs. X 60^ ^^^ ..
— Tu z — = 559 leet,
I lb. X 64.4
the distance traversed.
Again, th^ same result may be arrived at, according to Note i, by
multiplying the constant 64.4, in rule 4, for gravity, by the ratio of the
force and the weight, which in this case is j^, and 64.4 x j^ = 6.44.
Substituting 6.44 for 64.4 in that rule and formula, the formula becomes
Tj 60
^ = T — ^f — = 559 feet,
6.44 6.44
the distance traversed, as already found.
But the question may be answered more directly by the aid of the table
for falling bodies (No. 102, page 281). The height due to a velocity of
60 feet per second, is 55.9 feet; and it is to be multiplied by the inverse
ratio of the accelerating force and the weight of the body, or ^, or 10;
that is,
55.9 X 10 = 559 feet,
the distance traversed.
If the question be put otherwise — What space will a ball move over
before it comes to a state of rest, with an initial velocity of 60 feet per
GRAVITATION ON INCLINED PLANES. 285
second, allowing the friction to be iioth the weight of the ball? The
answer may be given, that the friction, which is the retarding force, being
iioth of the weight, that is of gravity, the space described will be 10 times
as much as is necessary for gravity, supposing the ball to be projected
vertically upwards to bring the ball to a state of rest. The height due to
the velocity is 55.9 feet; then
55.9 X 10 = 559 feet,
the space described by the ball.
Action of Gravity on Inclined Planes.
If a body freely descend an inclined plane by the force of gravity alone,
the velocity acquired by the body when it arrives at the foot of the plane,
is that which it would acquire by falling freely through the vertical height.
Or, the velocity is that " due " to the height of the plane.
The time occupied in making the descent is greater than that due to the
height, in the ratio of the length of the plane, or distance travelled, to
the height. The time is therefore directly in proportion to the length of
rhe plajie, when the height is the same.
The impelling or accelerating force by gravitation acting in a direction
parallel to the plane, is less than the weight of the body, in the ratio of the
height of the plane to its length. It is, therefore, inversely in proportion
to the length of the plane, when the height is the same.
The time of descent, under these conditions, is inversely in proportion
to the accelerating force.
If, for instance, the length of the plane be five times the height, the time
of making freely the descent on the plane by gravitation is five times
that in which a body would freely fall vertically through the height; and
the impelling force down the plane is '/j th of the weight of the body.
Problems on the descent of bodies on inclined planes are soluble by the
aid of the rules 7 to 14, for the relations of accelerating forces. But,
as a preliminary step, the accelerating force is to be determined, by
multiplying the weight of the descending body by the height of the plane,
and dividing the product by the length of the plane. For example, let a
body of 15 pounds weight gravitate freely down an inclined plane, the
length of which is fiver times the height, the accelerating force is 1 5 r 5 = 3
pounds. If the length of the plane be 100 feet, the height is 20 feet, and
the velocity acquired in falling freely firom the top to the bottom of the
plane would be, by rule 12,
z/ = 8/y/5^^ — 2z=s^ 20 =35.776 feet per second.
The time occupied in making the descent is, by rule 11,
^=}i V i^^^^^ = H\/ 500 = 559 seconds.
^ ' 3
Whereas, for a free vertical fall through the height, 20 feet, the time
would be, by rule 5,
t=%^ 20 = I.I 18 seconds,
which is '/s ^ ^^ ^^ ^^^^ ^^ making the descent on the inclined plane.
286 FUNDAMENTAL MECHANICAL PRINCIPLES.
Special Rules for the Descent on Inclined Planes.
The height and the length of an inclined plane may be substituted
for the accelerating force and the weight respectively in the rule (ii),
to find the time. Putting ^ = the height bf the plane, and /= the length
of the plane, the formula (ii)
'= Vk V ^ becomes t=}iy/ ^=% \/t,
/ "^ V A '* V A
/
°'''=i7=r (^5)
Rule 15. — Given the length and the height of the inclined plane, to find
the time in which a body would freely descend by gravitation. Divide the
length by four times the square root of the height; the quotient is the time
in seconds.
For example, the length of the plane is 100 feet, and the height is 20
feet, and the time is
100
^"47W =559 seconds,
as was found before.
Again, by inversion of the formula (15),
= 4/\/
h , and then
^=1^ (^^)
Rule 16. — Given the length of the inclined plane, and the time of
free descent by gravitation, to find the height through which the body
descends. Divide the square of the length by the square of the time
in seconds and by 16; the quotient is the length of the inclined plane.
For example, the length of the plane is 100 feet, and the time of descent
is 5.59 seconds; then the vertical height of the descent is
h = 5 = 20 feet, the height.
5.59^x16
Average Velocity of a Moving Body Uniformly Accelerated
OR Retarded.
The average velocity of a moving body uniformly accelerated or retarded,
during a given time or in a given space, is equal to half the sum of the
initial and final velocities; and if the body begin from a state of rest or
arrive at a state of rest, the average speed is half the final or initial velocity,
as the case may be. Thus, in the example of a ball rolling, the initial
speed or velocity is, in either case, 60 feet per second, and the terminal
speed is nothing; the average speed is therefore the same, namely, onehalf
of that, or 30 feet per second
MASS. — CENTRE OF GRAVITY. 287
MASS.
Weight IS not an essential property of a body; it is only the attraction of the
earth exerted upon the body. Suppose the attractive force to be suspended,
then the body would cease to have weight What would remain? Mass,
or substance, simply. But, though weight is not mass, it is a direct measure
of mass, in the same locality, or wherever the force of gravitation is the
same, for double the mass has twice the weight Weight alone, however,
is not sufficient as a universal measure of mass, since the weight of the
same mass would vary according to the force of gravitation for different
situations. The mass, therefore, varies inversely as the force of gravitation,
when the weight remains the same. That force is measurable by the height
through which a body falls in a given time, or by the velocity acquired at
the end of that time, say, a second, expressed by g. In its most general
form, then, the expression for the mass of a body comprises the weight
directly and the force of gravitation inversely; thus
^ = T (17)
in which m is the mass, w the weight, g the force of gravitation; that is to
say, the mass of. a body is equal to the weight of the body divided by the
force of gravity. Since the weight and the force of gravity vary in the
same ratio, the mass — of a body is the same at all places. As the quan
tity of matter of the same body is also constant whatever place it occupies,
the mass — gives an exact idea of the quantity of matter, and is a measure
of it
MECHANICAL CENTRES.
There are four mechanical centres of force in bodies, namely, the centre
of gravity, the centre of gyration, the centre of oscillation, and the centre
of percussion.
Centre of Gravity.
The centre of gravity is the physical centre of a body, or of a system of
bodies in rigid connection with each other, about which the gravitation of
the several particles of the body is selfbalanced, and at which it can be
freely suspended or supported in any position in a state of rest.
In various calculations, the whole weight or mass of a body is considered
as placed at its centre of gravity.
To find the centre of gravity of any plane figure mechanically: — Suspend
the figure by any point near its edge, and mark on it the direction of a
plumbline hung from that point; then suspend it from some other point,
and again mark the direction of the plumbline in like manner. Then the
centre of gravity of the surface will be at the point of intersection of the
two marks of the plumbline.
The centre of gravity of parallelsided objects may readily be found in
this way. For instance, to find the centre of gravity of the arch of a bridge;
draw the elevation upon paper to a scale, cut out the figure, and proceed
v/ith it as above directed, in order to find the position of the centre of
288 FUNDAMENTAL MECHANICAL PRINCIPLES.
gravity in elevation for the model. In the actual arch, the centre of gravity
will have the same relative position as in the paper model
In regular figures or solids the centre of gravity is the same as their
geometrical centres. Thus, the centre of gravity of a straight line, a
parallelogram, a prism, a cylinder, a circle, the circumference of a circle, a
ring, a sphere, and a regular polygon, is the geometrical centre of these
figures and solids respectively.
To find the centre of gravity of a triangle; draw a straight line from one
of its angles to the middle of the opposite side; the centre of gravity will be
in this line at a distance of twothirds of its length from the angle. Or,
draw a straight line from two of the angles to the middle of the opposite
sides respectively; the point of intersection of the two lines will be the
centre of gravity.
For a trapezium, or irregular foursided figure, draw the two diagonals,
and find the centres of gravity of each of the four triangles thus formed ;
then join each opposite pair of these centres of gravity. The joining lines
will cut each other in the centre of gravity of the figure.
For a cone and a pyramid, the centre of gravity is in the axis or centre
line, at a distance of threefourths of the length of the axis from the vertex,
or onefourth from the base.
For an arc of a circle, the centre of gravity lies in the radius bisecting the
arc, and the distance of it from the centre of the arc is found by multiplying
the radius by the chord of the arc, and dividing by the length of the arc; the
quotient is the distance of the centre of gravity from the centre of the circle.
For a sector of a circle, the centre of gravity is twothirds of the distance
of that of an arc, from the centre of the circle. It is found independently
by multiplying the radius by twice the chord of the arc, and dividing by
three times the length of the arc ; the quotient is the distance of the centre
of gravity from the centre of the circle.
For a parabolic space, the centre of gravity is in the axis, or centre line,
and its distance from the vertex is threefifths of the centre line or axis.
For a paraboloid, the centre of gravity is in the axis, at a distance from
the vertex of twothirds of the axis.
For two bodies, fixed or suspended one at each end of a straight bar, the
common centre of gravity is in the bar, at that point which divides the
distance between their individual centres of gravity, in the inverse ratio
of the weights respectively. For example, if two weights of 20 lbs. and
10 lbs. be suspended on a bar at a distance of 9 feet apart between their
centres of gravity, the common centre of gravity will divide the distance in
the ratio of i to 2, being 3 feet from the heavier weight, and 6 feet from
the lighter. In this example, the weight of the bar is neglected; but it may
be allowed for according to the following direction.
For more than two bodies connected in one system, the common centre
of gravity may.be found by finding, in the first place, the common centre
of gravity of two of them, and then finding that of these two jointly with
a third, and so on to the last body in the group.
Centre of Gyration. — Radius of Gyration. — Moment of Inertia.
The centre of gyration, or revolution, is that point in a revolving body, or
system of bodies, at a certain distance from tlie axis of motion, in which
the whole of the matter in revolution may, as an equivalent condition, be
CENTRE OF GYRATION. 289
conceived to be concentrated, just as If a pound of platinum were substituted
for a pound of revolving feathers, whilst the moment of inertia remains the
same. The work accumulated in the body, of which the moment of
inertia is a measure, remains in such a case the same, at the same angular
velocity; and, as a necessary consequence, if the same accelerating force
be applied to the body at the centre of gyration, as would actually be
expended over the distributed matter of the body to communicate to it its
angular velocity, the same angular velocity would be generated.
The distance of the centre of gyration from the axis of motion is called
the radius of g)rration ; and the moment of inertia is equal to the product
of the square of the radius of gyration by the mass or weight of the body.
The moment of inertia of a revolving body is found exactly by ascertain
ing the moments of inertia of every particle separately, and adding them
together; or, approximately, by adding together the moments of the small
parts arrived at by the subdivision of the body.
Rule i. To find the moment of inertia of a revolving body. Divide
the body into small parts of regular figure. Multiply the mass, or the
weight, of each part by the square of the distance of its centre of gravity
from the axis of revolution. The sum of the products is the moment of
inertia of the body.
Note. — ^The value of the moment of inertia obtained by this process
will be more nearly exact, the smaller and more numerous the parts into
which the body is divided.
Rule 2. To find the length of the radius of gyration of a body about a
given axis of revolution. Divide the moment of inertia of the body by its
mass, or its weight, and find the square root of the quotient. The square
root is the length of the radius of gyration; or
'/l ; (')
in which /// is the moment of inertia, and w is the weight of the body.
Note, — When the parts into which the body is divided are equal, the
radius of gyration may be determined by taking the mean of all the squares
of the distances of the parts from the axis of revolution, and finding the
square root of the mean square.
The following are useful examples of the radius of gyration of bodies: —
In a straight bar, or a thin rectangular plate, revolving about one of its
ends, the radius of gyration is equal to the length of the rod, multiplied by
^/
j^, or 0.5775.
In a straight bar, or a thin rectangular plate, revolving about its centre,
the radius of gyration is equal to half the length, multiplied by
y
y^y or 0.5775.
The general expression for the radius of g}'ration in a straight bar revolving
on any point of its length, is
in which a and b are the lengths of the two parts of the bar; that is to say,
19
290 FUNDAMENTAL MECHANICAL PRINCIPLES.
divide the sum of the cubes of the. two parts by three times the length of
the bar, and extract the square root of the quotient. The root thus found
is equal to the radius of gyration.
In a circular plate, a solid wheel of uniform thickness, or a solid cylinder
of any length, revolving on its axis, the radius of gyration is equal to the
radius of the object, multiplied by
y
J4, or 0.7071.
In a plane ring, like the rim of a flywheel, revolving on its axis, the radius
of gyration is approximately equal to the square root of half the sum of
the squares of the inside and outside radius of the rim.
In a thin circular plate, put in motion round one of its diameters,
the radius of gyration is equal to half the radius of the circle.
For the circumference of a circle, revolving about a diameter, the radius
of gyration is equal to the radius multiplied by 0.7071.
In the circumference of a circle revolving about its ovm. axis, the radius
of gyration is equal to the radius of the circle.
In a solid sphere revolving about a diameter, the radius of gyration is
equal to the radius multiplied by
V
V5, or 0.6324.
In the surface of a sphere, or an insensibly thin spherical shell, the
radius of gyration is equal to the radius multiplied by
^/
^, or 0.8615.
In a cone revolving about its axis, the radius of gyration is equal to the
radius multiplied by 0.1783.
Centre of Oscillation.
The centre of oscillation of a body vibrating about a fixed axis or centre
of suspension, by the action of gravity, is that point in which, if, as an
equivalent condition, the whole matter of the vibrating body were concen
trated, the body would continue to vibrate in the same time. It is the
resultant point of the whole vibrating energy, or of the action of gravity in
causing oscillation. As the particles of the body further from the centre of
suspension have greater velocity of vibration than those nearer to it, it is
apparent that the centre of oscillation is more distant than the centre of
gravity is from the axis of suspension, but it is situated in the centre line
which passes from the axis through the centre of gravity. It differs also
from the centre of gyration in this, that whilst the motion of oscillation is
produced by the gravity of the body, that of gyration is caused by some
other force acting at one place only.
The radius of oscillation, or the distance of the centre of oscillation from
the axis of suspension, is a third proportional to the distance of the centre
of gravity from the axis of suspension and the radius of gyration. Hence
the following rule for finding the radius of oscillation : —
CENTRE OF OSCILLATION. — THE PENDULUM. 29I
Rule 3. To find the radius of oscillation in a body vibrating on an
axis. Square the radius of gyration of the body, and divide by the distance
of the centre of gravity from the axis of suspension. The quotient is the
radius of oscillation. Or,
T» J r n ^ radius^ of gyration. / ^ v
Radius of oscillation = j. z f^ 7— p :— ( 3 ;
distance of centre of gravity from axis.
If the axis of suspension be in the vertex or uppermost point of a plane
figure, and the motion be edgewise, then.
In a right line, or straight rod, the radius of oscillation is twothirds
of the length of the rod.
In an isosceles, or equalsided triangle, it is threefourths of the height
of the triangle.
In a circle it is fiveeighths of the diameter.
In a parabola it is fivesevenths of the height.
But, if the oscillation of the plane figure be sidewise, then,
In a circle suspended at the circumference, the radius of oscillation is
threefourths of the diameter.
In a rectangle suspended by one of its angles, it is twothirds of the
diagonal.
In a parabola suspended by the vertex, it is fivesevenths of the axis
plus onethird of the parameter.
In a parabola suspended by the middle of its base, it is foursevenths
of the axis plus half the parameter.
In a sector of a circle suspended by the centre, it is threefourths of the
radius multiplied by the length of the arc, and divided by the length of the
chord.
In a cone it is fourfifths of the axis, plus the quotient obtained by
dividing the square of the radius of the base by five times the axis.
In a sphere it is twofifths of the square of the radius divided by the sum
of the radius and the length of the cord by which the sphere is suspended,
plus the radius and the length of the cord. For example, in a sphere
16 inches in diameter, suspended by a cord 25 inches long, the radius of
oscillation is
2x8^
h 8 I 25 = 0.78 + 33 = 33.78 inches.
5(8 + 25)
or 0.78 inch below the centre of the sphere.
It may be noted that the depression of the centre of oscillation below
the centre of the sphere, namely, 0.78 inch, is signified in the first quantity
in this equation.
77^ Pendulum.
A "simple pendulum" is the most elementary form of oscillating body, —
consisting theoretically of a heavy particle attached to one end of a cord,
or an inflexible rod, without weight, and caused to vibrate on an axis at
the other end, or the centre of suspension.
If an ordinary pendulum be inverted, so that the centre of oscillation shall
become the centre of suspension, then the first centre of suspension will
become the new centre of oscillation, and the pendulum will vibrate in the
292 FUNDAMENTAL MECHANICAL PRINCIPLES.
same time as before. This reciprocal action of the pendulum is a property
of all pendulous bodies, and it is known as the reciprocity of the pendulum.
The time of vibration of an ordinary pendulum depends on the angle or
the arc of vibration, and is greater when the arc of vibration is greater, but
in a very much smaller proportion; and if this arc do not exceed 4° or 5°,
that is to say, from 2° to 2^° on each side of the vertical line, the time of
vibration is sensibly the same, however the length of the arc may vary
within that limit. This property of a pendulum, of equal times of vibration,
is known as isochronism.
To construct a pendulum such that the time of vibration shall be the
same whatever the magnitude of the angle of vibration may be, it is neces
sary to cause the pendulum to vibrate, not in a circular arc, but in a
cycloidal curve. For this object the pendulum is suspended by a flexible
thread or rod, which oscillates between two cycloidal surfaces, on which it
alternately laps and unlaps itself; these are generated by a circle of which
the diameter is equal to half the length of the pendulum. By means of
the circle o b. Fig. 98, for example, of which the diameter is half the length
of the pendulum, describe the right and
left cycloidal curves oca, go' a', on the
horizontal line a a'; and draw the tangent
c B c', touching the cycloids at the middle
of their lengths. The halflengths o c, o c',
are equal to twice the diameter of the
generating circle OB, and consequently
equal to the length of the pendulum, which
Fig. 98.— Cycloidal Pendulum. will vibrate in cqual times, on the centre
of suspension o, between the entire half
lengths o c, o c', or in any shorter path. The curve c p c' thus described
by the pendulum, is itself a cycloidal curve, and is a duplicate of the other
cycloids. Though a cycloidal motion of the pendulum is necessary to render
it isochronous for all angles of vibration, yet taking very small arcs of the
cycloidal path on either side of the vertical line, they do not sensibly differ
from the circular arcs which would be described by an ordinary pendulum
of the same length (o p) swinging freely. Hence the reason that the ordinary
I>endulum vibrates in equal times when its vibrations do not exceed 4° or
5° in extent.
The length of the pendulum vibrating seconds at the level of the sea
in the latitude of London is 39.1393 inches, nearly a metre; at Paris
it is 39.1279; at Edinburgh it is 39.1555 inches; at New York, 39.10153
inches; at the equator it is 39.027 inches, and at the pole it is 39.197
inches. Generally, if the force of gravity, or the length of the seconds
pendulum at the equator be represented by i, the gravity, or the length of
pendulum at other latitudes will be as follows : —
Length of Seconds Pendulum,
At the equator i. 00000
„ 30'' latitude 1.00141
„ 45 » 1.00283
» 52 „ 1.00357
,, 60 „ 1.00423
„ 90 „ (the pole) 1.00567
THE PENDULUM. 293
According to these ratios, the force of gravity, and the length of the
seconds pendulum, at the pole, are Viyeth greater than at the equator; there
being a difference of length of between a fourth and a fifth of an inch.
The following are the relations of the lengths of pendulums and the times
of their vibrations, that is to say, of such as vibrate through equal angles,
or of which the total angle of vibration does not exceed 4° or 5°: —
The times of vibration of pendulums are proportional to the square root
of the lengths of the pendulums.
Conversely, the lengths oT pendulums are to each other as the squares of
the times of one vibration, or inversely as the squares of the numbers of
vibrations in a given time.
The length of the seconds pendulum at London, 39.1393 inches, may
be taken as a datum for calculation applicable to pendulums of different
lengths, and to different times of vibration.
Rule 4. To find the time of vibration of a pendulum of a given
length. Divide the square root of the given length in inches by the
square root of 39.1393, or 6.2561. The quotient is the time of a vibration
in seconds. Or
^ 391393 6.2561'
in which / is the given length of pendulum in inches, and / the time of
vibration in seconds.
Rule 5. To find the number of vibrations per second of a pendulum of
given length. Divide 6.2561 by the square root of the length in inches.
The quotient is the number of vibrations per second.
For the number of vibrations per minute. Divide 375.366 by the
square root of the length in inches. The quotient is the number of
vibrations per minute. Or
« = ^^^ (per second); ( 5 )
n = ^l^^ (^rmmute); (5)
in which n is the number of vibrations.
Rule 6. To find the length of a pendulum when the time of a
vibration is given. Multiply the square of the time of one vibration in
seconds by 39.1393. The product is the length of the pendulum in
inches. Or
/=/2x39.i393 (6)
Rule 7. To find the length of a pendulum when the number of
vibrations per second is given. Divide 39.1393 by the square of the num
ber of vibrations in a second. The quotient is the length of the pendulum
in inches.
When the number of vibrations per minute is given. Divide 140,900
by the square of the number of vibrations in a minute. The quotient is
the length of the pendulum in inches. Or
/, 391393 /.)
«^ (per second)'
/_ 140,900 (7)
n^ (per minute)
294 FUNDAMENTAL MECHANICAL PRINCIFl.ES.
A pendulum may be shorteneci and yet vibrate in the same time as
before, by the action of a second weight fixed on the pendulum rod above
the centre of suspension. Here the upper weight counteracts the lower,
and there is only the balance of gravitating force due to the preponderance
of the lower weight available for vibrating both masses. The mass being
thus increased while the gravitating force is diminished, a longer time is
required for each vibration when the length of pendulum remains unaltered,
or the pendulum may be shortened so that the time of the vibrations con
tinues the same. By varying the height of the upper weight above, the
centre of suspension, and thus varying the level of the common centre of
gravity, the period of vibration is varied in proportion.
Rule 8. To find the weight of the upper bob of a compound pendulum
necessary to vibrate seconds, when the weight of the lower bob is given,
and the respective distances of the bobs from the centre of suspension.
Multiply the distance in inches of the lower bob from the centre of suspen
sion by 39.1393, and from the product subtract the square of that distance
(i); again, multiply the distance in inches of the upper bob from the
centre of suspension by 39.1393, and add the square of that distance
(2); multiply the lower weight by the remainder (i), and divide by the
sum (2). The quotient is the weight of the upper bob. Or
^ = w 99i393xD) D^ (8)
(391393 >"^ + <^'
in which D and d are the respective distances of the lower and upper bobs
from the centre of suspension, and W, a/, their respective weights.
Thus, by means of a second bob, pendulums of small dimensions
may be made to vibrate as slowly as may be desired. The metronome,
an instrument for marking the time of music, is constructed on this
principle, the upper weight being slid and adjusted on a graduated rod
to measure fast or slow movements.
The Centre of Percussion.
If a blow is struck by an oscillating or revolving body moving about a
fixed centre, the percussive action is the same as if the whole mass of
the body were concentrated at the centre of oscillation. That is to say,
the centre of percussion is identical with the centre of oscillation, and
its position is found by the same rules as for the centre of oscillation.
If an external body is so struck that the mean line of resistance passes
through the centre of percussion, then the whole force of percussion is
transmitted directly to the external body; on the contrary, if the revolving
body be struck at the centre of percussion, the motion of the revolving
body will be absolutely destroyed, so that the body shall not incline either
way, just as if every particle separately had been struck.
CENTRAL FORCES.
When a body revolves on an axis, every particle moves in a circle of
revolution, but would, if freed, move off in»a straight line, forming a tangent
to the circle. The force required to prevent the body or partiple flying
from the centre is called cctiiripetal force, and the tendency to fly from
the centre is centrifugal force. These forces are equal and opposite —
examples of action and reaction — and are classed as central forces.
CENTRAL FORCES. 295
Centrifugal force varies as the square of the speed of revolution.
It varies as the radius of the circle of revolution.
It varies as the mass or the weight of the revolving body.
Let c be the centrifugal force, w the weight of the revolving body,
r the radius of revolution or gyration, m the mass of the body = , in which
^ = 32.2 or gravity; and v the linear or circumferential velocity; then
m v^ w v^
c—  =
r 32.2 r
That is to say, the centrifugal force of a revolving body is equal to the
weight of the body multiplied by the square of the linear velocity, divided
by 32.2 times the radius of gyration.
If the height due to the velocity be substituted for the velocity in the
above equation, the height // being equal to  — , then
64.4
2WV^ 2 w h
€= = ,
64.4 r r
and
c \ w \ \ 2 h \ K
That is to say, the centrifugal force is to the weight of the body as twice the
height due to the velocity is to the radius of gyration.
From the first equation the following rules for revolving bodies are
deduced, for finding one of the four elements when the other threie are
given: — namely, the centrifugal force, the radius of gyration, the linear
velocity, and the weight.
Rule i. For the centrifugal force. Multiply the weight by the square
of the speed, and divide by 32.2 times the radius of gyration. The quotient
is the centrifiigal force. Or
c^ (i)
32.2 r
Rule 2. For the linear velocity. Multiply the centrifugal force by the
radius of gyration, and by 32.2, and divide by the weight; and find the
square root of the quotient. The root is the velocity. Or
/x2.2cr / V
^ w
Rule 3. For the weight Multiply the centrifugal force by the radius of
gyration, and by 32.2, and divide by the square of the velocity. The
quotient is the weight. Or
32.2 c r / V
'^ = ^:jr : ; (3)
Rule 4. For the radius of gyration. Multiply the weight by the square
of the velocity, and divide by the centrifugal force, and by 32.2. The
quotient is the radius of gyration. Or
w iP'
r= (4)
32.2 c
296 FUNDAMENTAL MECHANICAL PRINCIPLES.
Note, — ^When the velocity is expressed as angular velocity, in revolutions
per unit of time, it is to be reduced to linear or circumferential velocity by
multiplying it by the radius of gyration and by 6.28; or
f = 6.28 if r,
in which 7/ is the angular velocity.
By substitution and reduction in equation (i), the following equation in
terms of the angular velocity is arrived at : —
0.8165 c=wr7/^, /j)
from which is found
'=5176i = '"S«"^' (6)
That is to say, the centrifugal force is equal to the weight multiplied by the
radius of gyration and by the square of the angular velocity, and by 1.225.
MECHANICAL ELEMENTS.
The function of mechanism is to receive, concentrate, diffuse, and apply
power to overcome resistance. The combinations of mechanism are num
berless; but the primary elements are only two, namely, the lever and the
inclined plane. By the lever, power is transmitted by circular or angular
action; that is to say, by action about an axis; by the inclined plane, it is
transmitted by rectilineal action. The principle of the lever is the basis of
the pulley and the wheel and axle; that of the inclined plane is the basis of
the wedge and the screw.
For the present, frictional resistance and the weight of the mechanism
are not considered; the terminal resistance is called the weight; and the
elemental mechanisms are to be treated as in a state of equilibrium, in
which the power exactly balances the weight without actual movement.
The action, or work done, will be subsequently treated.
The Lever.
The elementary lever is an inflexible straight bar, turning on an axis or
fixed point, called the fulcrum; the force being transmitted by angular
motion about the fulcrum, from the
point where the power is applied to the
point where the weight is raised, or
other resistance overcome. There are
three varieties of the lever, according
as the fulcrum, the weight, or the
power is placed between the other two,
but the action is, in every case, re
Pjg. 99— Lever. duciblc to that of three parallel forces
in equilibrium (page 275).
First. The power is applied at one end ^, of the lever ab c. Fig. 99, and
transmitted through the fulcrum, ^, to the weight at the other end c. The
moments of the power and the weight about the fulcrum are equal, or
power y^ab weight y>b c.
That is, the product of the power by its distance from the fulcrum is equal
THE LEVER.
297
to the product of the weight by its distance from the fulctum. Conse
quently
power : weight : : b c \ ab^
that is, the power and the weight are to each other inversely as their
respective distances from the fulcrum.
The ratio of the length of the power end of the lever to the length of the
weight end is called the leverage of the power. The respective lengths,
Fig. 99, being 7 feet and i foot, the leverage is 7 to i, or 7,
The three varieties of the lever are
grouped together in Figs. 100, 10 1, and 1
102. In each case, the lever is supposed I _, I
to be 8 feet long and divided into feet. J ^ f \ , — p 1 . ^.
The leverage, in the first, is 7 to i, or 7; ^^  "S
in the second, 8 to i, or 8; in the third, 'f^
yi to I, or ^ : showing that, in the first Q I A
case, the power balances seven times its yr ^
own amount; in the second case, eigh
times its amount; in the third case, only
Fig. xoo. — Lever, ist kind.
^ +
R
T ' ' r
Ji
a
II''''
IT
Fig. loi. — Lever, ad kind.
Fig. loa. — Lever, 3d kind.
oneeighth of itself, because it is nearer to the fulcrum than the weight.
In each case the moments of the power and the weight about the fulcrum
are equal, for, in each case.
Pxdf ^ = Wx3^.
{a)
The pressures exerted at the extremities of the lever act in the same
direction, and the sum of them is equal and opposite to the intermediate
pressure, whether it be that of the fulcrum, the weight, or the power ( — ).
From this the pressure on the fulcrum may be found. If it be in the
middle, the pressure is equal to the sum of the power and the weight, that
is, 60 + 420 = 480 lbs. in the example above ; if at one end, it is equal to
the difference of them, that is, it is 480 — 60 = 420 lbs. when the weight is
in the middle, and it is 607^^ = 52)^ lbs. when the power is in the
middle.
From the equation for the equality of moments,
orPxL =Wx/, {b)
in which L and / are the respective distances of the power and the weight
from the fulcrum, rules may be formed for finding any one of the four
quantities, when the other three are given.
Rule i. To find the power. Multiply the weight by its distance from
the fulcrum, and divide by the distance of the power from the fulcrum.
The quotient is the power.
298 FUNDAMENTAL MECHANICAL PRINCIPLES.
Or, divide the weight by the leverage. The quotient is the power. Or
Rule 2. To find the weight Multiply the power by its distance firom
the fulcrum, and divide by the distance of the weight from the fulcrum.
The quotient is the weight.
Or, multiply the power by the leverage. The product is the weight Or
w=y (2)
Rule 3. To find the distance of the power from the fulcrum. Multiply
the weight by its distance from the fulcrum, and divide by the power. The
quotient is the distance of the power from the fulcrum. Or
^ = 17 (3>
Rule 4. To find the distance of the weight from the fulcrum. Multiply
the power by its distance from the fulcrum, and divide by the weight The
quotient is the distance of the weight from the fulcrum. Or
9 If the weight of the lever be included in such calculations, its influence
is the same as if its whole weight or its mass were collected at its centre of
gravity. Thus, if the lever of the first kind, Fig. 100, weighs 30 lbs., and its
centre of gravity be at the middle of its length, the weight of the lever
cooperates with the power, at a mean distance of 3 feet from the fulcrum.
By equality of moments
(P X 7) X (30 X 3) = W X I = 420 lbs. X I,
and P X 7 = 420  90 = 330 lbs.;
therefore P, the power at the end of the lever required to balance the
Fig. Z03. — Inclined Lever.
Fig. xo4.^Inclined Lever.
weight, is only 33047 = 47.1 lbs. in cooperation with the weight of the
lever, as compared with 60 lbs., without reckoning the aid from this source.
When the lever is inclined to the direction of the forces, as in Fig. 103,
THE LEVER.
299
equilibrium, or the equality of moments, may nevertheless be maintained.
Drawing the horizontal line a' b d through the fulcrum, to meet the ver
ticals through the power and the weight at a! and ^, the moments of the
power and the weight are to be estimated on the horizontal lengths ci b, b c'\
and
the moment V %a! b the moment W xb i/.
The equality of moments may be proved in another way. Let the
power and the weight be resolved, in order to find the pressures on the
ends of the lever, at right angles to it, and thus to arrive at the moments
as estimated on the actual length of the lever. Let the verticals through
the ends of the lever, a m and cn^ Fig. 104, represent the power and the
weight respectively, and draw a Y and c W perpendicular to the lever, and
/// P' and n W parallel to if, completing the triangles a m P', c n W. Then
a P' and c W are the components of the power and the weight respectively
tending to turn the lever; and, it may be added, they bear the same ratio
to each other as the power and the weight Consequently, if these com
ponents be multiplied by the respective lengths of the lever, the products
will be the moments of the components, and the moments will be equal; or
the moment ^ P' x df ^ = the moment cW xb c.
These two methods of analyzing and finding the moments of the forces
acting on an inclined lever — one, combining a reduced length of lever witjj
the whole power and weight; the other, combining the whole length of
lever with a reduced power and weight — lead to one conclusion, that a
lever, if balanced in one position, is balanced in every other position,
when the forces continue to act in parallel lines.
■®
O
Fig. 105.— Bent Lever.
Fig. X06. — Bent Lever.
The conditions of equilibrium in a bent lever may be defined sinriilarly.
Let the lever a b c. Fig. 105, be bent at the fulcrum b; draw the horizontal
line of b d, then the moments of the power and the weight are reckoned
on the lines a' b, b c*, and they are equal to each other; or
300
FUNDAMENTAL MECHANICAL PRINCIPLES.
»' «•
Again, let the forces acting on a lever, whether straight or bent, be
otherwise than vertical or parallel. When the arms of the lever are at
right angles, and the power and the weight applied at right angles to the
arms, as in Fig. io6, the moments are reckoned directly on the arms, ab,bc^
as in a straight lever; and
the moment P x « ^ ^ the moment W x ^ r.
The thrust, or pressure on the fulcrum, is in this case less than the sum of
the power and the weight; and it may be determined by constructing a
parallelogram upon the two arms of the
lever, the arms representing inversely the
respective forces. That is, a b represents
the magnitude and direction of the weight
W, and b c th®se of the power P. The
diagonal b y, of the parallelogram repre
sents the magnitude and direction of the
third force acting at the fulcrum to oppose
O^ ^ the other two and maintain equilibrium.
When the same lever is tilted into an
IT oblique position, the power continuing to
Fig. 107.— Bent Lever. act horizontally on the lever, Fig. 107,
draw the vertical b' d through the end c of
the lever, and produce the power line ap \.o meet it at }/, Complete the
parallelogram a! 1/ (f b; then the sides a* b and b d zi^ the perpendiculars
to the directions to the power and weight, on which the moments are
reckoned, so that
the moment P x a' ^ = the moment W x ^ ^.
The diagonal ^ ^ is the resultant force at the fulcrum.
^
^
6
Fig. X08. — Bent I.,ever.
Fl?. T09. — Serpentine Lever.
If the power do not act horizontally, but in some other direction, a /,
Fig. 108, produce the powerline pa and draw ba^ perpendicular to it.
THE LEVER. 3OI
Draw b (f 2& before ; then the moments are reckoned on the perpendiculars
b (fy b c\ and, as before,
To find the resultant force at the fulcrum. On the fulcrum ^ as a centre
describe arcs of circles with the radii b a' and b (f, and draw b a^y b (f
respectively parallel to the directions of the weight and the power, to cut
the arcs at (f and c'^. Complete the parallelogram, and the diagonal b b^
represents in magnitude and direction the resultant force at the fulcrum.
In this solution the power and the weight are assumed to act exactly,
or sensibly, in the same plane.
Again, in the serpentine lever a b c^ Fig. 109, supposed to be a pump
handle, the power P is applied obliquely in the direction a P. Produce
P a and W c, and draw tlie perpendiculars b af^b <f from the fulcrum for
the lengths of the moments, then
Pxrt'^ = Wx^^.
Construct the parallelogram, as in the foregoing figure, and the diagonal
b b" represents the resultant force at the
fulcrum. ^/
By similar treatment the action of the . :'" ^^^^^"^.
forces in levers of the second and third :
kinds may be analyzed. The lever of the ®'~'^,"yf^^ Av
second kind, a c by Fig. 1 10, in an oblique \ \^^^^ C/
position, is acted on horizontally by the : * P
power and the weight at a and c; draw /'^
the vertical b d a!y then b c' and b of are \^
the distances at which the forces act from W
the fulcrum, or the lengths of the mo Fig. no.— Lever of the ad kind.
ments, and
and the horizontal resultant force at the fulcrum is equal to the difference
of the weight and the power.
If more than two forces be applied to a lever in a state of equilibrium,
the sum of the moments tending to turn tlie lever in one direction is equal
to the sum of those tending in the opposite direction.
If two or more levers are connected consecutively one to the other, so
that they act as one system, with the power and the weight at the extremi
ties, then, in equilibrium, the ratio of the power to the weight is the product
of the separate inverse ratios of all the levers. For example, in a connected
series of three levers, having each their arms in the ratio of 2 to i, the
combined inverse ratio is found by multiplying 2 by 2 and by 2 ; thus
first lever 2 to i ratio,
second lever 2 to i ratio,
third lever 2 to i ratio,
compound ratio 8 to i.
That is; the power is to the weight as i to 8.
302
FUNDAMENTAL MECHANICAL PRINCIPLES.
The Pulley.
The pulley is a wheel over which a cord, or chain, or band is passed, in
order to transmit the force applied to the cord in another direction. It is
equivalent to a continuous series of levers, with equal arms on one fulcrum
or axis, and affords a continuous circular motion instead of the intermittent
circular motion of one lever. The weight W, Fig. iii, is sustained by the
power P, by means of a cord passed over the pulley A, in fixed supports,
and the centre line abc represents the element of the lever, from the ends
of which the power and the weight may be conceived to depend, turning
on the fulcrum b. By equality of moments, V y.ab = Vf y.bc; and the
arms or radii a b, b c being equaJ, the power is equal to the weight, and the
counterpressure at the fulcrum is equal to twice the weight.
When the power and weight act in directions #t an angle with each other,
as in Fig. 112, the acting radii ab, be, representing the element of a bent
^^V1lh^Vll»^^^vftf^^^1^l
s>x.v^ 7vA.*»a>vvvv •■ .,.^.^^
•vr
Fig. xxL—Pullcy.
Fig. 112.— Pulley.
Fig. 1x3. —Pulley.
lever, are lines drawn from the centre perpendicular to the directions of the
power and weight The power is equal to the weight, but the counter
pressure on the fulcrum is less than twice the weight, and is represented by
the diagonal b V of the parallelogram formed by the
radii bc^, bd, drawn from the fulcrum parallel to
the directions of the power and the weight respec
tively.
Another construction for the parallelogram of
forces in the action of the pulley is obtained by
producing the directions of the power and the weight
beyond the pulley. Fig. 113, intersecting each other
at y, then forming the parallelogram, and drawing
the diagonal b' if 2& the resultant pressure on the
fulcrum.
Thus the single fixed pulley acts like a lever of the
first kind, and simply changes the direction of force,
without modifying the intensity of the power.
But the pulley may be employed as a lever of the
second kind by suspending the weight to the axis
of the pulley, and fixing one end. of the cord to a point as a fulcrum
point. Thus, in Fig. 114, the weight W is suspended from the axis c.
Fig. 114. — Movable Pulley,
as a lever of the 3d kind.
THE PULLEY. $0$
the cord is fixed to the point i', and the power P acts through the diameter
acd, in which ^ is the fulcrum. By equality of moments,
that is, the product of the power by the diameter of the pulley is equal to
the product of the weight by the radius of the pulley,
and the leverage being as 2 to 1, the power is only
half the weight.
In acting as a lever of the third kind, the power is
applied to the axis a. Fig. 115, one end of the cord
b«ing fixed at ^, and the weight attached at the other
end, c. In this case, by equality of moments the
product of the power by Jhe radius of the pulley is
equal to that of the weight by the diameter, and the
leverage being as i to z, the power is twice the weight
These demonstrations with respect to movable
pulleys only apply to cases of parallel cords; that is
to say, when the direction of the power is parallel to J,ie»trofihe dkini"
that of the weight. If, on the contrary, they be inclined '
to each other, as in Fig. 116, in which the weight is suspended by the
axis, the power becomes greater than half the weight, as is shown by the
parallelogram of which the diagonal c'tf represents the weight, and the
sides e'y, i (f, the pull on the fulcrum, and the power exerted to sustain
the weight Each of these sides is greater than half the diagonal.
Fig. ii«.MovaMe PuBey. Fir. iij.FullcyBladci.
ComMrta/ions of PuUeys. — Fast and Loose Pulleys. — In these last two
applications of the pulley, it becomes movable when in action, and by com
bining two or more movable pulleys on the same or different axles in one
block, with one cord, the gain of power may be increased in the same pro
portion. The movable block A, Fig. 117, carrying the weight, is used
304 FUNDAMENTAL MECHANICAL PRINCIPLES,
with a fixed counterpart E, the rope is attached by one end to the fixed
block, and is passed over the movable and fixed pulleys, from one to Che
other in succession, the power being applied to the other end, as at P.
'I'his system is known as &st and loose pulieyb locks.
The fixed end of the rope is sometimes attached to the movable block.
Rule i. To find the power necessary to balance a weight or resistance
by means of a system of fast and loose pulleys. Divide the weight by the
is carried; that is, the number of ropes which
alock. The quotient is the power required to
2 rope is attached to the fixed block, the num
Ti the loose block is twice the number of mov
may be found by dividing the weight by twice
:ys. •
is attached to the movable block, the divisor
imber of movable pulleys plus i.
ber of movable pulleys; if the fixed end of the
block,
I: (.)
ipe be attached to the movable block,
^ (..)
ight or resistance that will be balanced by a
system of fast and loose pulleys. Multiply the
r by the number of ropes proceeding from the
Lble block. The product is the required weight
■, when the rope is attached to the fixed block,
ply the power by twice the number of movable
ys.
■, when the rope is attached to the movable
:, multiply the power by twice the number of
Lble pulleys plus i.
, in the first case,
W2»P; (2)
t: second case,
w.(»«+.)P M
;ain, a combination may be formed of a num
f movable pulleys, as in Fig. 118, each of which,
C, is suspended by a cord, with one end fixed
e roof and the other end fixed to the axis of
ext pulley. The weight W is hung to the axis
delivers half the weight to the second pulley B,
veight hanging to it, or onefourth of the first
ey C; from which only oneeighth of the first
de or neutral pulley D lo the power P. In
THE PULLEY.
30s
general the divisor for the power is 2", or the //th power of 2, n being the
number of movable pulleys.
Rule 3. To find the power necessary to balance a weight by means of a
system of separate movable pulleys, with separate cords consecutively con
nected as above described. Divide the weight by that power of 2 of which
the index is the number of movable pulleys. The cjuotient is the power or
force required to balance the weight
Or, divide and subdivide the weight successively by 2 as many times as
there are movable pulleys to find the power required. Or
P =
W
(3)
Rule 4. To find the weight that can be balanced by a given power, by
means of a system of separate movable pulleys as above described. Mul
tiply the power by that power of 2 of which the index is the number of
movable pulleys. The product is the weight required.
Or, multiply the power successively by 2 as many times as there are
pulleys. Or
W=PX2'' (4)
Note. — It is necessary that the cords should be parallel to each other,
as in the illustration, in order that the above rules, 3 and 4, may apply.
Wheel and Axle.
The wheel and axle may be likened to a couple of pulleys of different
diameters united together on one axis, of which the larger, a, Fig. 1 19, is the
wheel, and the smaller, c, the axle, with a common ful
crum, b', the centre line abc representing the elements of
a lever. The weight W on the axle at c is balanced *by
the power P, on the wheel at a. The moments are equal,
or
Pxfl!^ = Wx^r;
and the power is to the weight inversely as their distances
from the centre; or
V :^Y :\ be : ab.
Fig. 119.— Wheel
ana Axle.
If a crank handle be substituted for the wheel, making
a windlass, the radius of the crank is substituted for that
of the wheel in estimating the ratio of the forces.
Of the four elements, namely, the radius of the wheel or crank, the radius
of the axle or roller, the power, and the weight, if three be given, the fourth
can be found as follows, putting R and r for the respective radii.
Rule i. To find the power. Multiply the weight by the radius of the
axle, and divide by the radius of the wheel. The quotient is the power.
Or
WxJ
(i)
Rule 2. To find the weight Multiply the power by the radius of the
20
3o6
FUNDAMENTAL MECHANICAL PRINCIPLES.
wheel, and divide by the radius of the axle. The quotient is the weight
Or
W = Px —
r
(»)
Rule 3. To find the radius of the wheel. Multiply the weight by the
radius of the axle, and divide by the power. The quotient is the radius of
the wheel. Or
R=^^ (3)
Rule 4. To find the radius of the axle. Multiply the power by the
radius of the wheel,* and divide by the weight. The quotient is the radius
of the axle. Or
PR / V
(4)
r
W
Note, — The diameters of the wheel and the axle or roller may be
employed in the calculations instead of the radii.
Inclined Plane.
The inclined plane is a slope, or a flat surface inclined to the horizon, on
which weights may be raised. By such substitution of a sloping path for
a direct vertical line of ascent, a given weight can be raised by a power
which is less than the weight itself.
There are three elements of calculation in the inclined plane: — the plane
itself, A B, Fig. 120; the base, or horizontal length, AC; and the height or
vertical rise B C ; together forming a right
angled triangle. The weight W is to be
raised through a height equal to C B, and
for that object is drawn up the slope from
A to B. It is partly supported during the
O^^^^v^ (^^)^^ ascent, and it is in virtue of this degree of
^'*"^^\i^7/ support given to the weight that such a
^dFC^ "dead pull" as that of a direct vertical
lift is avoided, and that it can be raised
by a power much less than its o\mi weight.
Let the weight W be kept at rest on the
incline by the power P, acting in the line b P', parallel to the plane. Draw
the vertical line ^^ to represent the weight; also bV perpendicular to the
plane, and complete the parallelogram V c. Then the vertical weight ba
is equivalent to b b\ which is the measure of support given by the plane to
the weight, and b c, which is the force of gravity tending to draw the weight
down the plane. The power required to maintain the weight in equilibrium
is represented by this force be. Thus, the power and the weight are in
the ratio of beta b a.
Since the triangle of forces abe is similar to the triangle of the incline
A B C, the latter may be substituted for the former in determining the
relative magnitude of the forces, and
P : W :: ^^ : fl^ :: BC : AB,
Fig. X30.Incluied Plane.
THE INCLINED PLANE.
307
that is, the power, acting parallel to the inclined plane, is to the weight, as
the height of the plane to its length. Then, by equality of moments,
PxAB = WxBC,
or P X length of inclined plane = W x height of inclined plane (a)
For example, take the length of the inclined plane, 24 feet; the height,
2 feet; and the weight to be raised, 360 lbs. The power required to
balance the weight is equal to 360 x 2 h 24 = 30 lbs.
Again, the base, A C, of the inclined plane, represents the element of
the pressure of the weight on the inclined plane.
It is thus seen that the sides of the triangle formed by an inclined plane,
its base, and its height, are respectively proportional as follows : —
The inclined plane to the weight at rest on the plane.
The base to the pressure of the weight on the plane.
The height to the power acting parallel to the plane.
When the power acts in a direction parallel to the base, as in Fig. 121, in
which the power P supports the weight
W in the direction d V\ parallel to the
base; draw the vertical da to represent
the weight, and the line d^ perpen
dicular to the incline, and complete the
parallelogram d' c. The weight If a, de
composed, is equivalent to ^ ^', the per
pendicular to the incline, representing
the pressure of the weight upon the
plane, and d c, the force of traction, or
the power P. Here the pressure ^ ^ on
the plane is greater than the v/eight, and
the power ^ ^ is greater than when the line of traction is parallel to the
incline.
The triangles adc, ABC, being similar, the ratios of the power and the
weight are as follows : —
F :W :: dc : ad :: BC : AC; {3)
that is, they are to each other as the height of the plane to its base; and
the inclined plane, the base, and the height, are respectively proportional
as follows: —
The inclined plane to the pressure of the weight on the plane.
The base to the weight at rest on the plane.
The height to the power acting parallel to the base.
If the power be applied in any direction above that which is parallel to
the incline, though the pressure of the weight on the plane will be less than
the weight itself, yet, as in the previous case, the power is greater than is
necessary when it acts in a direction parallel to the plane. Thus, in
Fig. 122, in which the power P acts at a divergent angle in the direction
d P', draw the vertical da, the perpendicular dd\ to the plane, and the
extension of the power line to c, and complete the parallelogram. Then,
the weight is represented by d a, the pressure on the incline by d V, and the
power by a ^ or ^ ^.
Fig. lax. — Inclined Plane.
3o8
FUNDAMENTAL MECHANICAL PRINCIPLES.
Fig. laa.— Inclined Plane.
For comparison, the parallelogram that would represent the relative
forces arising from a power acting parallel to the plane, is added on the
figure in dotted lines extending to the angles b" and (f. It shows that the
pressure on the plane is greater
than when the power is di
vergent, but that the power
is less.
It follows that the longer
the inclined plane, when the
height is the same, the less is
the power required to balance
the weight; in fact, the power
simply varies in the inverse
ratio of the length of the
plane.
If two inclines, A B and
B D, of unequal lengths and
the same height, be united
back to back on the line BC,
then two weights, W and W,
on the respective planes, connected by a cord over a pulley at the summit
B, will balance each other, when they are in the ratio of the lengths of the
planes on which they rest That is,
W : W : : A B : B D.
From the formula ( a ), rules may be formed for finding one of the following
four elements when the other three are given, namely, the length of the
inclined plane, the height of
it, the weight, and the power
to balance the weight when
acting in a direction parallel
to the incline.
Rule i. To find the power.
Multiply the weight by the
height of the plane, and divide
by the length. The quotient
is the power.
Rule 2. To find the weight Multiply the power by the length of the
plane, and divide by the height. The quotient is the weight
Rule 3. To find the height of the inclined plane. Multiply the power
by the length, and divide by the weight The quotient is the height
Rule 4. To find the length of the inclined plane. Multiply the weight
by the height of the plane> and divide by the power. The quotient is the
length.
Identity of the Inclined Plane and the Lever,
Though the inclined plane is distinguished from the lever in the mode of
operation, inasmuch as there is no motion about a mechanical centre, as in
the lever, yet the conditions of equilibrium on the inclined plane may be
established on the principle of the lever. Suppose a round weight W kept
at rest on the incline A B by a power P parallel to the incline, passing
Fig. 123. — Double Inclined Plane.
LEVERAGE ON THE INCLINED PLANE.
309
through the centre a. Draw ab perpendicular to the incline; the point b
is the point of contact of the weight with the incline. Draw the vertical
line a d, and the perpendicular ^ ^ to it. Then the lines ab, be form a bent
lever a be, of which b is the fulcrum, and
ab,be the arms. The weight acts at the
extremity e of the short arm, and the power
at the extremity a of the long arm; and the
power and the weight are to each other
inversely as the relative arms of the lever,
ab,be. Now, as abe and A B C are similar
triangles, the arms a b, be axe to each other
as the length and the height A B, B C, of
the incline, and
'P : W : : b e : a b : : B C : A B:
Fig. 124. — Leverage ofh an Inclined Plane*.
that is, the power is to the weight as the height of
length, which is the proportion already established
The ratio of the length of an inclined plane to
the leverage of the plane, and the products of the
the plane, and of the weight into the height of the
moments of the power and the weight.
Suppose, again, that the power is applied at P,
a P, passed round and over the weight parallel
the inclined plane to its
( « ) page 307).
its height may be called
power into the length of
plane, may represent the
Fig. 125, through a cord
to the incline; then the
C A
Fig. 125. — Leverage on an Inclined Plane.
Fig. I a6.— Wedge.
diameter of the weight a b becomes the long arm of the lever a be, through
which the power acts, being double the length of the arm a b, Fig. 1 24,
where the power is applied at the centre of the weight. By thus doubling
the leverage, the power is halved, and the ratio of the power to the weight
is as half the height of the plane to its length.
In this case there is the action of a movable pulley combined with an
inclined plane; the rolling weight moved by a cord lapped round it, repre
senting a movable pulley with the weight attached to the axle. Thus the
leverage of the power on the inclined plane can be doubled.
The Wedge.
The wedge is a pair of inclined planes united by their bases, or " back to
back," as A B C B , Fig. 1 26. Whereas inclined planes are fixed, wedges are
moved, and in the direction of the centre line C A, against a resistance
equally acted on by both planes of the wedge. The function of the wedge
310 FUNDAMENTAL MECHANICAL PRINCIPLES.
is to separate two bodies by force, or divide into two a single body. In
some cases the w^edge is moved by blows, as in splitting timber; in others
it is moved by pressure. The action by simple pressure is now to be con
sidered.
The pressure P is applied to a wedge at the head B B' at right angles to
the surface, and the resistance or "weight" to be overcome is opposed to
the wedge and acts at right angles to the faces A B, A B', at the middle
points of which, a, a, it is supposed, to be concentrated. Whilst the wedge
and the power move through a space equal to the length of the wedge A C,
the weight is moved or overcome through a space equal to the breadth of
the wedge B B'; and, as the power is to the weight inversely as the spaces
described, they are to each other directly as the breadth to the length of
the wedge. That is,
P : W : : B B' : A C,
and the product of the power by the length of the wedge is equal to the
product of the weight by the breadth of the wedge; or
PxAC = WxBB;
or P X length = W X breadth of wedge {c)
By the aid of the parallelogram the same conclusions are arrived at
Thus, in Fig. 126, produce the directions of the two resistances, W«, \N a, to
meet in the middle of the wedge at by complete the parallelogram, and draw
the diagonals aca and bb\ The diagonal b b' is the resultant of the two
forces ab,ab, and represents the pressure on the head of the wedge. Again,
in the triangle a be, whilst a b represents, in magnitude and direction, the
perpendicular pressure of the resistance on the wedge, a c, which is perpen
dicular to the centre line of the wedge, represents, in magnitude and
direction, the force applied in overcoming the resistance. The ratio of the
power to the weight is therefore as bb' to a c. And, as the triangle abb' is
similar to the triangle ABB',
P : W :: ^^ : ^z^ :: BB' : AC;
that is, the power is to the weight as the breadth of the wedge to its length.
From the formula ( c ), the following rules for wedges acting under pres
sure, as distinct from impact, are deduced : —
Rule i. To find the weight or transverse resistance. Multiply the
power by the length of the wedge, and divide by the breadth of the head.
The quotient is the weight.
Rule 2. To find the power. Multiply the weight or transverse resistance
by the breadth of the head, and divide by the length of the wedge. The
quotient is the power.
Rule 3. To find the length of the wedge. Multiply the weight by the
breadth of the wedge, and divide by the power. The quotient is the
length of the wedge.
Rule 4. To find the breadth of the wedge. Multiply the power by the
length of the wedge, and divide by the weight. The quotient is the breadth
of the wedge.
Note, — I. The length of the wedge is taken as the distance from the
head to the point of intersection of the sides.
THE SCREW.
311
2. The power may be applied at the point of the wedge by drawing,
instead of at the head by pressing.
3. The power may be applied in a direction parallel to one of the sides
of the wedge, and the relation of the power to the weight may be found by
construction, in the same manner as for the inclined plane, when the power
is applied in a direction parallel to the base. See proportion ( ^ ), page 307.
The Screw.
The screw is an inclined plane lapped round a cylinder. Take, for
example, an inclined plane ABC, Fig. 127, and bend it into a circular form,
resting on its base, Fig. 128, so that the ends meet The incline may be
Fig. 127.
continued winding upwards round the same axis, and thus winding or
helical inclined planes of any required length and height may be con
structed. The helix thus arrived at being placed upon a solid cylinder,
and the dead parts of the helix removed, the product is an ordinary screw.
The inclined fillet is the " thread " of the screw, and the screw is called
"external." But the thread may also be applied
within a hollow cylinder, and then it is " internal,"
such as an ordinary " nut " is.
The distance of two consecutive coils apart,
measured from centre to centre, or from upper side
to upper side, — ^literally the height of the inclined
plane, — ^for one revolution, is Qie "pitch" of the
screw.
The effect of a screw is estimated in terms of the
pitch and the radius of the handle employed to turn either it or the nut,
one on the other; and the leverage of the power is the ratio of the circum
ference of the circle described by the power end of the handle to the pitch.
The radius is to be measured to the central point where the power is
applied.
The circumference being equal to the radius multiplied by twice 3. 14 16,
or 6.28,
Fig. ia8.
P : W :: / : /X6.28,
in which / is the pitch and r the radius; also
6.28 Pr=Wx/;
{d)
that is, 6.28 times the product of the power by the radius of the handle is
equal to the product of the weight by the pitch. Whence the following
rules relative to the power of a screw, for finding any one of those four
quantities when the other three are given : —
Rule i. To find the power. Multiply the weight by the pitch, and
312 FUNDAMENTAL MECHANICAL PRINCIPLES.
divide by the radius of the handle and by 6.28. The quotient is the
power. Or
P = ^^ (X)
6.28 r ^^
Rule 2. To find the weight. Multiply the power by the radius and by
6.28, and divide by the pitch. The quotient is the weight. Or
^^6,28^r ^^j
Rule 3. To find the pitch. Multiply the power by the radius of the
handle and by 6.28, and divide by the weight. The quotient is the pitch.
Oi
^ 6.28 Pr , V
/=— w~ : ^^)
Rule 4. To find the radial length of the handle. Multiply the weight
by the pitch, and divide by the power and by 6.28. The quotient is the
length of the handle. Or
"j5^p <^)
JYofe. — When the power is applied through a wheel fixed to the screw,
the acting diameter of the wheel may be substituted for the radius in the
above rules and formulas, and the constant becomes 3.14.
Similarly, should the powerwheel be fixed to the nut so as to turn
it upon the screw, instead of the screw within the nut, the same sub
stitutions may be made.
WORK.
Work consists of the sustained exertion of pressure through space.
The English unit of work is one footpound; that is, a pressure of one
pound exerted through a space of one foot.
The French unit of work is one kilogrammetre; that is, a pressure of one
kilogramme exerted through a space of one metre.
One kilogrammetre is equal to 7.233 footpounds.
In the performance of work by means of mechanism, the work done
upon the weight is equal to the work done by the power. This prin
ciple of the equality of work is deducible from the principle of the
equality of moments, and is expressed generally by the equation
PxH = Wx>4, (a)
in which H is the height or space moved through by the power, and
// the height or space moved through by the weight at the same time.
It signifies that the product of the power by the space through which it has
acted is equal to the product of the weight by the space through which
it has acted.
Again,
P : W : : /i : H,
signifying that the power is to the weight inversely as the respective heights
or spaces moved through by them in the same time.
WORK. — WORK WITH THE MECHANICAL ELEMENTS. 313
Work done with the Lever.
On the principle of the equality of moments, the power and the weight
in the lever, neglecting frictional resistance, are to each other inversely as
the lengths of the arms upon which they act, that is, of their radii of
motion; and inversely as the arcs or spaces passed through or described by
the ends of the arms. If the weighted lever, Fig. 99, page 296, be moved
by the power, the power descends through an arc at a, and the weight is
raised through an arc at c. These arcs may be taken as the heights moved
through, and are proportional to the lengths of the respective arms, ab^b c.
In this example, these are as 7 to i, and if the power descend 7 inches the
weight is raised only i inch; but the weight raised is seven times the power
applied, and "what is gained in power is lost in speed," or, more correctly,
in space moved through. The equality of work thus developed from the
equality of moments is thus expressed
power X arc a — weight x arc c («)
To show this arithmetically, let the weight be raised through i foot; then,
with a leverage of 7 to i, the power descends 7 feet, and taking it, as before,
at 60 lbs., the weight it raises will be 60 lbs. x 7 = 420 lbs., and the equation
of work is
60 lbs. X 7 feet = 420 lbs. x i foot,
(or 420 footpounds) (or 420 footpounds).
Again,
power : weight : : arc c : arc a^
expressing the principle of virtual velocities, the relative velocities being
indicated by the arcs «, c.
Work done with the Pulley.
In using the single fixed pulley. Fig. iii, page 302, the power is equal to
the weight, and the spaces through which they move in the same time are
equal.
With the movable pulley, Fig. 114, the weight is suspended at the axle,
and in raising the weight i foot, the power at the circumference, with a
leverage of 2, passes through 2 feet and is only half the weight If
P and W be 20 lbs. and 40 lbs. respectively, the equality of work is thus
expressed —
(P) 20 lbs. X 2 feet = (W) 40 lbs. x i foot = 40 footpounds;
and by means of this pulley a weight double the power is raised half the
height through which the power is applied.
Conversely, when the weight is suspended at the circumference of the
movable pulley, Fig. 115, and the power applied at the axle, the leverage
is J^ ; the power is therefore double the weight, and moves through i foot
whilst the weight moves through 2 feet. Thus
(P) 40 lbs. X I foot = (W) 20 lbs. X 2 feet = 40 footpounds.
In a system of fast and loose pulley blocks, Fig. 117, page 303, the
power being equal to the weight divided by the number of ropes, then, by
314 FUNDAMENTAL MECHANICAL PRINCIPLES.
equality of work, the space through which the power is moved is equal to
the height through which the weight is raised, multiplied by the number of
ropes. Suppose that there are three movable pulleys and six ropes; if the
weight, izo lbs., be raised i foot) each rope is shortened i foot and the
power is moved 6 feet And
(P) 20 lbs. X 6 feet = (W) 120 lbs. x i foot = 120 footpounds.
Work done with the Wheel and Axle.
'heel, Fig. iig, page 305, makes one revolution, the axle also
The power descends or traverses a space equal to the cir
thewheel = 2 (ai) x 3.i4i6,whilst the weight is raised through
to the circumference of the axle = 2 (i^:) x 3.1416. If the
wheel be r foot 6 inches, and that of the axle 3 inches, the
s are 9.42 feet and 1.57 feet, being as 6 to i; and the power
t, conversely, are as 1 to 6. If the power be 20 lbs., then
;P) 20 lbs. X g.42 feet = {W) rao lbs. x 1.57 feet.
(188.4 footpounds) (188.4 footpounds).
Work done with the Inclined Plane.
is raised in opposition to gravity, and the work done on it is
the product of the weight into the vertical height of the
;. The work done by the power is enpressed by the product
into the length of the'plane, These two products express
es of work, and
Px/=Wx^,
nated at (a), page 307, to express equality of moments.
e, the length of the plane is 24 feet and the height 2 feetj
1,20 lbs., the power 10 lbs. Then, the work done in rising
the whole of the incline is 240 lbs., thus
(P) 10 lbs. X 24 feet = (W) 1 20 lbs. x 2 feet
(240 footpounds) {240 footpounds).
lere supposed to be applied in a direction parallel to the plane.
a direction at an angle to the plane, as in Fig. 122, page 308,
lolved into its components, parallel and perpendicular to the
the line ^ <;' parallel to the incline; then the power applied,
;nt to the force actually expended & 1!, and to the pressure
)n c c". The consumption of power is expressed by the pro
irallel equivalent, b c, into the length of tiie plane. Taking,
LS above, the weight, 120 lbs., and the active power, 10 lbs.,
ly the parallel force b tf; then the amount of the horizontal
)ower applied, b c, is found by proportion, thus
AC:ABi:i/:if;
arallel and horizontal forces are to each other as the base to
the incline.
WORK IN MOVING BODIES. 315
Work done with the Wedge.
Supposing the wedge driven by a constant pressure through a distance
equal to its length, the work done by the power is expressed by the power
into the length, and the work done on the weight is expressed by the pro
duct of the weight into the breadth of the wedge. By equality of work,
PxL = WxB,
as before stated, in expressing equality of moments.
If the wedge be driven for only a part of its length, the work done by
the power is in the proportion of the part of the length driven; and the
work done on the weight is similarly in the proportion of the part of the
breadth by which the resisting surfaces are separated.
Work done with the Screw.
In one revolution of the screw, the weight is raised through a height
equal to the pitch of the thread, whilst the power acts through the circum
ference of the circle described by the point at which it is applied to a lever.
The products of the power and the weight by the spaces (^escribed by
them are equal, or
Px6.28r = Wx/,
as before stated (page 311) to express equality of moments.
Work done by Gravity.
The work done by gravity on a falling body is equal to the weight of the
body multiplied by the height through which it falls.
Work accumulated in Moving Bodies.
The quantity of work stored in a body in motion is the same as that
which would be accumulated in it by gravity if it fell from such a height as
would be sufficient to give it the same velocity; in short, from the height
due to the velocity. (See Graviit, page 2^^),^ The accumulated work
expressed in footpounds, is equal to the height so found in feet, multiplied
by the weight of the body in pounds. The height due to the velocity is
equal to the square of the velocity divided by 64.4, and the work and the
velocity may be found directly from each other, according to the following
rules: —
Rule i. Given the weight and velocity of a moving body, to find the
work accumulated in it. Multiply the weight in pounds by the square of
the velocity in feet per second, and divide by 64.4. The quotient is the
accumulated work in footpounds.
Or, putting U for the work, v for the velocity, and 7v for the weight,
U = 4^ (I)
64.4
Or, secondly: — Multiply the weight in pounds by the height in feet due
to the velocity. The product is the accumulated work in footpounds. Or,
putting A for the height,
U = «/x// ( i^)
3l6 FUNDAMENTAL MECHANICAL PRINCIPLES.
Work done by Percussive Force.
If a wedge be driven by blows or strokes of a hammer or other heavy
mass, the effect of the percussive force is measured by the quantity of work
accumulated in the striking body. This work is calculated by the preceding
rules, from the weight of the body and the velocity with which the blow is
delivered, or directly from the height of the fall, if gravity be the motive
power.
The useful work done through the wedge is equal to the work delivered
upon the wedge, supposing that there is no elastic or vibrating reaction
from the blow, just as if the work had been delivered by a constant pres
sure equal to the weight of the striking body, exerted through a space equal
to the height of the fall, or the height due to its final velocity.
Of course, in order to give effect to the constant pressure on the wedge,
now imagined to be brought into action, the pressure would require to be
applied to the resisting medium through some combination of the mechanical
elements.
But where elastic action intervenes, a portion of the work delivered is
uselessly absorbed in elastically straining the resisting body; and the elastic
action may be, in some situations, so excessive as to absorb the whole of
the work delivered. In this case, there would not be any useful work done.
These remarks, applied to the action of a blow on a wedge, are applicable
equally to the action of a blow of the monkey of a piledriver upon a pile.
If there be no elastic action, the work delivered being the product of the
weight of the monkey by the height* of its fall, is equal to the work done in
sinking the pile; that is, to the product of the frictional and other resistance
to its descent by the depth through which it descends for one blow of the
monkey.
Supposing that the pile rests upon and is absolutely resisted by a hard
unyielding obstacle, the work done becomes wholly useless, and consists of
elastic or vibrating action ; or it may be that the head of the pile is split
open.
HEAT.
THERMOMETERS.
The action of Thermometers is based on the change of volume to which
bodies are subject with a change of temperature, and they serve, as their
name implies, to measure temperature. Thermometers are filled with air,
water, or mercury. Mercurial thermometers are the most convenient, because
the most compact. They consist of a stem or tube of glass, formed with a
bulbous expansion at the foot to contain the mercury, which expands into
the tube. The stem being uniform in bore, and the apparent expansion of
mercury in the tube being equal for equal increments of temperature, it
follows that if the scale be graduated with, equal intervals, these will indi
cate equal increments of temperature. A sufficient quantity of mercury
having been introduced, it is boiled to expel air and moisture, and the tube
is hermetically sealed. The freezing and the boiling points on the scale
are then determined respectively by immersing the thermometer in melting
ice and afterwards in the steam of water boiling under the mean atmospheric
pressure, 14.7 lbs. per square inch, and marking the two heights of the
column of mercury in the tube. The interval between these two points is
divided into 180 degrees for Fahrenheit's scale, or 100 degrees for the
Centigrade scale, and degrees of the same interval are continued above and
below the standard points as far as may be necessary. It is to be noted
that any inequalities in the bore of the glass must be allowed for by an
adaptation of the lengths of the graduations. The rate of expansion of
mercury is not strictly constant, but increases ¥dth the temperature, though,
as already referred to, this irregularity is more or less nearly compensated
by the varying rates of expansion of glass.
In the Fahrenheit Thermometer, used in Britain and America, the number
0° on the scale corresponds to the greatest degree of cold that could be
artificially produced when the thermometer was originally introduced. 32°
("the freezingpoint") corresponds to the temperature of melting ice, and
212® to the temperature of pure boiling water — in both cases under the
ordinary atmospheric pressure of 14.7 lbs. per square inch. Each division
of the thermometer represents i® Fahrenheit, and between 32** and 212°
there are i8o^
In the Centigrade Thermometer, used in France and in most other
countries in Europe, o** corresponds to melting ice, and 100® to boiling
water. From the freezing to the boiling point there are 100°.
In the R^umur Thermometer, used in Russia, Sweden, Turkey, and
Egypt, o** corresponds to melting ice, and 80° to boiling water. From the
freezing to the boiling point there are 80°.
3 1 8 HEAT.
Each degree Fahrenheit is  of a degree Centigrade, and y of a degree
Reaumur, and the relations between the temperatures indicated by the
different thermometers are as follows : —
C. = I (F. 32). R. = I (F. 32). C. = f R.
G. being the temperature in degrees Centigrade.
R. do. do. Reaumur.
F. do. do. Fahrenheit.
That is to say, that Centigrade temperatures are converted into Fahrenheit
temperatures by multiplying the former by 9 and dividing by 5, and adding
32° to the quotient; and conversely, Fahrenheit temperatures are converted
into Centigrade by deducting 32°, and taking ths of the remainder.
Reaumur degrees are multiplied by  to convert them into the equivalent
Centigrade degrees; conversely, ths of the number of Centigrade degrees
give their equivalent in Reaumur degrees.
Fahrenheit is converted into Reaumur by deducting 32° and taking ths
of the remainder, and Reaumur into Fahrenheit by multiplying by f , and
adding 32® to the product
Tables No. 104, 105 contain equivalent temperatures in degrees Centigrade
for given degrees Fahrenheit, from 0° F., or zero on the Falirenheit scale, to
608° F. ; and conversely, the temperature in degrees Fahrenheit correspond
ing to degrees Centigrade, from 0° C, or zero on the Centigrade scale, to
320° C.
EQUIVALENT TEMPERATURES.
319
Table Na 104. — Equivalent Temperatures by the Fahrenheit
AND Centigrade Thermometers.
Degrees
Degrees
Degrees
Degrees
Degrees
Degrees
Degrees
Fahr.
Centigrade.
Fahr.
Centigrade.
Fahr.
Centigrade.
Fahr.
Centigrade.
17.78
+ 38
+ 3.34
+ 76
+ 24.45
+ 114
+ 45.56
+ I
17.23
39
3.90
77
25.00
1^5
46.11
2
16.67
40
4.45
78
25.56
116
46.67
3
16.II
41
5.00
79
26.12
117
47.23
4
15.56
42
5.56
80
26.67
118
47.78
5
15.00
43
6. 1 1
81
I
27.23
119
48.34
6
14.45
44
6.67
82
27.78
120
48.90
7
13.90
45
7.23
^3
28.34
121
49.45
8
13.34
46
7.78
84
28.89
122
50.00
9
12.78
47
8.34
85
29.45
123
50.56
10
12.23
48
8.89
86
30.00
124
5 I.I I
II
11.67
49
945
87
30.55
125
5167
12
II. II
so
10.00
88
31. II
126
52.23
13
10.56
51
10.56
89
31.67
127
52.78
14
10.00
52
II. II
90
32.22
128
5334
15
9.45
S3
11.67
91
32.78
129
53.90
16 '
8.89
54
12.23
92
3333
130
54.45
17
8.34
55
12.78
93
33.89
131
5500
18
7.78
56
13.34
94
34.45
132
5556
19
7.23
57
13.90
95
35.00
133
56.11
20
6.67
58
14.45
96
35.56
134
56.67
21
6. 1 1
59
15.00
97
36.11
135
57.23
22
556
60
15.56
98
36.67
136
57.78
23
5.00
61
16.11
99
37.23
137
58.34
24
4.45
63
16.67
100
37.78
138
58.90
25
3.90
63
17.23
lOI
38.34
139
59.45
26
3.34
64
17.78
102
38.90
140
60.00
27
2.78
65
18.34
103
39.45
141
60.56
28
2.23
66
18.89
104
40.00
142
61. II
29
1.67
67
19.45
105
40.56
143
61.67
30
I. II
68
20.00
106
41. II
144
62.23
31
0.56
69
20.56 j
107
41.67
145
62.78
32
0.00
70
21. II
108
42.23
146
63.34
33
+ 0.56
71
21.67
109
42.78
147
63.90
34
I. II
72
22.23
no
43.34
148
64.45
35
1.67
73
22.78 ,
III
43.90
149
65.00
36
2.23
74
2334
112
44.45
150
65.56
37
2.78
75
23.90
"3
45.00
151
66.11
HEAT.
Table No. 104 {continued).
■^fX"
Demes
C«l«rade.
^^
Deerea
^S^."
Cenugrade.
f5^
Dcgr«s
Cenugndt.
66.67
+ 193
+ 89.45
+ 234
■HI 12.23
+ 275
+ 13500
67^3
194
90.00
235
112.78
376
I35.56
67.78
195
90.56
236
"334
277
136"
68.34
196
91. II
237
113.90
278
136.67
6S.90
197
91.67
238
"445
279
13723
6945
198
9223
239
115.00
380
13778
70.00
"99
93.78
240
115.56
381
138.34
70.56
300
9334
241
116.11
383
138.90
71.11
201
9390
242
116.67
2S3
139.45
71.67
Z02
9445
243
11723
284
140.00
73.33
203
95O0
344
117.78
285
140.56
73.78
304
95.56
245
118.34
386
141. II
7334
205
96.11
246
118.90
287
141.67
739°
206
96.27
247
"945
288
142.23
7445
207
9723
248
120.00
389
142.78
7500
208
97.78
249
120.56
390
143.34
7556
209
9834
250
121.11
291
13390
76.1.
98.90
25"
131.67
292
144.45
76.67
211
994S
252
132.23
293
I4500
77^3
312
100.00
253
122.78
394
14556
77.78
213
100.S6
254
'23.34
295
146. 1.
78.34
214
lOI.II
12390
296
146.67
78.90
315
101.67
256
12445
397
147.23
7945
zi6
102.23
257
125.00
398
147.78
217
102.78
258
125.56
299
148.34
80.56
3l8
10334
259
136.11
300
148.90
81. II
219
103.90
260
126.67
301
14945
81.67
10445
261
12723
302
150.00
83.23
221
105.00
262
137.78
303
15056
83.78
232
105.56
363
128.34
304
151.11
8334
223
106. 1 1
264
128.90
305
151.67
83.90
224
106.67
26s
129.45
306
152.23
84.4s
22s
107.23
266
130.00
307
152.78
85.00
336
107.78
267
130.56
308
153.34
85.56
227
108.83
268
13111
309
'53.90
86.11
328
108.90
369
13167
310
15445
86.67
229
109.45
270
132.23
3"
155.00
8723
230
371
13278
312
J5S56
87.78
231
"a?6
272
13334
313
88.34
23a
iii.ii
273
13390
314
Ise'eS
88.90
"33
111.67
274
13445
3«S
15723
EQUIVALENT TEMPERATURES.
321
Table No. 104 {continued).
Fahrenheit and Centigrade.
D^re«s
De^ees
Degrees
Decrees
Degrees
De^ees
Degrees
Fahr.
Degrees
Fahr,
Centigrade.
Fahr.
Centigrade.
Fahr.
Cenugrade.
Centigrade.
+ 316
+ 157.78
+ 357
+ 180.56
+ 398
+ 203.34
+ 439
+ 226.11
317
15^34
358
181. II
399
203.90
440
226.67
318
158.90
359
181.67
400
204.45
441
227.23
319
159.45
360
182.23
401
205.00
442
227.78
320
160.00
361
182.78
402
205.56
443
228.34
321
160.56
362
183.34
403
206.11
444
228.90
322
161. II
363
183.90
404
206.67
445
229.45
323
161.67
364
184.45
405
207.23
446
230.00
324
162.23
365
185.00
406
207.78
447
230.56
325
162.78
366
185.56
407
208.34
448
231. II
326
163.34
367
186. 1 1
408
208.90
449
231.67
327
163.90
368
186.67
409
209.45
45^
232.23
328
164.45
369
187.23
410
210.00
451
232.78
329
165.00
370
187.78
411
210.56
452
233.34
330
165.56
371
188.34
412
211. II
453
233.90
331
166.II
372
188.90
413
211.67
454
234.45
332
166.67
373
189.45
414
212.23
455
235.00
333
167.23
'374
190.00
415
212.78
456
235.56
334
167.78
375
190.56
416
213.34
457
236.11
335
168.34
376
191. II
417
213.90
458
236.67
336
168.90
377
191.67
418
214.45
459
237.23
337
169.45
378
192.23
419
215.00
460
237.78
338
170.00
379
192.78
420
215.56
461
238.34
339
170.56
380
193.34
421
2l6.II
462
238.90
340
171. II
381
193.90
422
216.67
463
239.45
341
171.67
382
194.45
423
217.23
464
240.00
342
172.23
383
195.00
424
217.78
465
240.56
343
172.78*
384
195.56
425
218.34
466
241. II
344
173.34
385
I96.II
426
218.90
467
241.67
345
173.90
386
196.67
427
219.45
468
242.23
346
174.45
387
197.23
428
220.00
469
242.78
347
175.00
388
197.78
429
220.56
470
243.34
348
• 175.56
389
198.34
430
221. II
471
243.90
349
1 76. 1 1
390
198.90
431
221.67
472
244.45
350
176.67
391
199.45
432
222.23
473
245.00
351
177.23
392
200.00
433
222.78
474
245.56
352
177.78
393
200.56
434
223.34
475
246.11
353
178.34
394
201. II
435
223.90
476
246.67
354
178.90
395
201.67
436
224.45
477
247.23
355
179.45
396
202.23
437
225.00
478
247.78
356
>
180.00
397
202.78
438
225.56
479
248.34
21
Table No. 104 (cotUitnud).
'ahrenmeit and Ckntigrao
Ccnli^ijdc
DegMS
Centigrade,
■iX"
Degrees
Cemigrade.
'^^
C™.«™ic,
+ 248.90
+ 5"3
+ 267.23
+ 546
+ 285.56
+ 579
+ 30390
24945
SI4
26778
547
286.11
5B0
30445
250.00
515
26834
548
286.67
58'
305.00
250.56
516
26890
549
287.23
582
3055''
25I"
5"7
269.45
55°
287.78
583
306.11
2S,.67
5>8
270.00
55'
288.34
584
306.67
252.23
5>9
270.56
552
288.90
585
30723
252.78
520
271.11
553
289.45
586
307.78
253.34
521
271.67
554
290.00
S87
308.34
25390
522
272.23
555
290.56
588
308.90
25445
523
272.78
556
291. II
589
30945
255.00
524
27334
557
291.67
590
310.00
255.56
525
27390
558
292.23
591
310.56
256.,.
526
27445
559
292.78
592
311. II
256.67
527
275.00
560
29334
593
311.67
25723
528
2755'
561
293.90
594
312.23
257.78
529
2761.
562
294.45
595
312.78
25834
530
276.67
563
295.00
596
31334
25890
531
27723
564
29556
597
31390
25945
532
27778
565
296.11
598
31445
26000
533
278.34
566
296.67
599
31500
26056
534
278.90
567
297.23
600
31556
26111
535
279.45
568
297.78
601
316.11
261.67
536
280.00
569
298.34
602
316.67
262.23
537
280.56
570
298.90
603
31723
262.78
538
281.11
571
29945
604
31778
263.34
539
281.67
572
300.00
605
3T8.34
263.90
540
282.23
573
300.56
606
318.90
264.45
541
282.78
574
301. 11
607
31945
265.00
542
28334
575
301.67
608
320.00
26556
543
28390
576
302.23
266.11
544
28445
577
302.78
366.67
545
285.00
578
303.34
EQUIVALENT TEMPERATURES.
323
Table No. 105. — Equivalent Temperatures by the Centigrade and
Fahrenheit Thermometers.
Degrees
Degrees
Degrees
Degrees
Fahr.
Degrees
Degrees
Degrees
Degrees
Cent.
Fahr.
CenL
Cent.
Fahr.
Cent.
Fahr.
20
 4.0
+ 21
+ 69.8
+ 62
+ 1436
+ 103
+ 217.4
19
2.2
22
71.6
63
145.4
104
219.2
18
0.4
23
734
64
147.2
105
22l!o
17
+ 1.4
24
75.2
65
149.0
106
222.8
16
3.2
^5
77.0
66
150.8
107
224.6
15
5.0
26
78.8
67
152.6
108
226.4
H
6.8
27
80.6
68
154.4
109
228.2
13
8.6
28
82.4
69
156.2
no
230.0
12
10.4
29
84.2
70
158.0
III
231.8
II
12.2
30
86.0
71
159.8
112
233.6
10
14.0
31
87.8
72
161.6
"3
2354
9
15.8
32
89.6
73
163.4
114
237.2
8
17.6
33
91.4
74
165.2
"5
239.0
7
19.4
34
932
75
167.0
116
240.8
6
21.2
35
95.0
76
168.8
117
242.6
5
23.0
36
96.8
77
170.6
118
2444
4
24.8
^l
98.6
78
172.4
119
246.2
3
26.6
38
100.4
79
174.2
120
248.0
2
28.4
39
102.2
80
176.0
121
249.8
I
30.2
40
104.0
81
177.8
122
251.6
32.0
41
105.8
82
179.6
123
253.4
+ I
33.8
42
107.6
83
181.4
124
255.2
2
35.6
43
109.4
84
183.2
125
257.0
3
37.4
44
III. 2
85
185.0
126
258.8
4
392
45
1 130
86
186.8
127
260.6
5
41.0
46
1 14.8
87
188.6
128
262.4
6
42.8
47
1 16.6
88
190.4
129
264.2
7
44.6
48
1 18.4
89
192.2
130
266.0
8
46.4
49
120.2
90
194.0
131
267.8
9
48.2
50
122.0
91.
195.8
132
269.6
10
50.0
51
123.8
92
197.6
133
271.4
II
51.8
52
125.6
93
199.4
134
273.2
12
53.6
53
127.4
94
201.2
135
275.0
13
554
54
129.2
95
203.0
136
276.8
14
57.2
55
1310
96
204.8
137
278.6
15
59.0
56
132.8
97
206.6
138
280.4
16
60.8
57
134.6
98
*
208.4
139
282.2
17
62.6
58
136.4
99
210.2
140
284.0
18
64.4
59
138.2
100
212.0
141
285.8
19
66.2
60
140.0
lOI
213.8
142
287.6
20
68.0
61
141.8
102
215.6
143
289.4
324
HEAT.
Table No. 105 {continued).
Centigrade and Fahrenheit.
Depves
Jjcgtccs
Degrees
Degrees
Degrees.
DcgFMS
Cent.
Fahr.
CenL
Fahr.
C«lL
Fahr.
Cent.
Fahr.
+ 144
+ 291.2
+ 189
+ 372.2
1
+ 234
+ 453.2
+ 279
+ 5342
H5
293.0
190
. 374.0
235
455.0
280
536.0
146
294.8
191
3758
236
456.8
281
537.8
147
296.6
192
377.6
237
458.6
282
539.6
148
298.4
193
379.4
238
460.4
283
541.4
149
300.2
194
381.2
239
462.2
284
543.2
150
302.0
195
383.0
240
464.0
285
545.0
151
3038
196
384.8
. 241
465.8
286
546.8
152
305.6
'97
386.6
242
467.6
287
548.6
153
307.4
198
388.4
243
469.4
288
1
5504
154
309.2
199
390.2
244
471.2
i 289
552.2
'55
31 10
200
392.0
245
473.0
I 290
554.0
156
312.8
201
393.8
246
474.8
291
555.8
157
314.6
202
3956
247
476.6
292
557.6
158
316.4
203
397.4
248
478.4
293
559.4
159
318.2
204
3992
249
480.2
: 294
561.2
160
320.0
205
401.0
250
482.0
i 295
563.0
161
321.8
206
402.8
251
483.8
> 296
564.8
162
3236
207
404.6
252
485.6
297
566.6
163
325.4
208
406.4
253
487.4
298
568.4
164
327.2
209
408.2
254
489.2
299
570.2
'^l
329.0
210
410.0
255
491.0
300
572.0
166
330.8
211
41 1.8
256
492.8
301
573.8
167
332.6
212
413.6
257
494.6
302
575.6
168
3344
213
4154
258
496.4
303
577.4
169
336.2
' 214
417.2
259
498.2
304
579.2
170
338.0
215
419.0
260
500.0
305
581.0
171
3398
216
420.8
261
501.8
306
582.8
172
341.6
217
422.6
262
503.6
307
584.6
173
343.4
218
424.4
263
505.4
308
586.4
174
345.2
219
426.2
264
507.2
309
588.2
175
347.0
220
428.0
265
509.0
310
590.0
176
348.8
221
429.8
266
510.8
311
591.8
177
350.6
222
4316
267
512.6
312
593.6
178
352.4
223
433.4
268
514.4
313
5954
179
354.2 '
224
435.2
269
516.2
314
597.2
180
356.0
225
437.0
270
518.0
315
599.0
181
357.8
226
438.8
271
519.8
316
600.8
182
359.6
227
440.6
272
521.6
317
602.6
183
361.4
228
442.4
273
523.4
318
604.4
184
363.2
229
444.2
274
525.2
319
606.2
185
365.0
230
446.0
275
527.0
320
608.0
186
366.8
231
447.8
276
528.8
187
368.6
232
449.6
277
530.6
188
370.4
233
451.4
278
532.4
AIRTHERMOMETERS.
32s
C
t
AirThermometers.
Airthermometers, or gasthermometers, though inconvenient because
bulky, are, by reason of the great expansiveness of air, superior to such as
depend upon the expansion of liquids or solids, in point of delicacy and
exactness. In any thermometer, whether liquid or gas, the indications
depend jointiy upon the expansion by heat of the fluid substance, and that
of the tube which holds it The expansion of mercury is scarcely seven
times that of the glass tube within which it expands, and the exactness of
its indications are interfered with by the variation in the expansiveness of
glass of different qualities. In the gasthermometer, on the contrary, the
expansiveness of the gas is 160 times that of the glass, and the inequalities
of the glass do not sensibly affect the indications of the instrument
Gasthermometers, or, as they are commonly called, airthermometers,
are designed either to maintain a constant pressure with a varying volume
of air, or to maintain a constant volume of air while the pressure varies.
In the first case. Fig. 119, the thermometer consists of a reservoir a, to be
placed in the substance of which the temperature
is to be ascertained; a tube d/, connected at a
suitable distance by a small tube alf to the reservoir;
a tube cd, open above, through which mercury is
introduced into the instrument; a stopcock r to
open or close a communication — ist, between the
tube dfzjoA the atmosphere; 2d, between the base
of the tube cd and the atmosphere; 3d, between
the two tubes df^ cd\ 4th, between both these
tubes and the atmosphere. The tube df^ which is
carefully gauged, answers the purpose of the gradu
ated tube of the mercurythermometer, and receives
the air driven over by expansion from the reservoir,
at the same time that it is maintained at or near
the temperature of the surrounding atmosphere.
Thus the air is divided between the reservoir a and
the tube df, of which the air in the former is at the ^* "9.AirThcnnomcter.
temperature of the substance under observation, and that in the latter is
at the temperature of the atmosphere. These two portions of air support
the same pressure, which qm at all times be approximated to that of the
atmosphere by means of the cock r, through which the mercury is allowed
to escape until it arrives at the same level in the two tubes. By means of
a formula embracing the respective volumes of the two portions of air and
the temperature of the atmosphere, the temperature of the substance under
observation is determined. But it is apparent that, when applied as a
pyrometer to the measurement of high temperatures — ^higher, that is to
say, than the boiling point of mercury (676° F.) — ^the greater part of the air
passes by expansion into the tube df^ leaving but a small remainder in the
reservoir a. A serious objection to this is that the proportion of air which
passes over into the tube df for a new increase of temperature is very
small, and is with difficulty measured with sufficient precision.
The second form of air thermometer, in which the pressure varies whilst
the volume remains the same, was used by M. Regnault in his researches.
326
HEAT.
f
The temperature is measured by means of the increased elastic force of the
inclosed air, and the instrument is both more convenient and more precise
than that in which the volume varies, for at all temperatures the sensibility
of the instrument is the same. At high temperatures the apparatus is liable
to distortion under the pressure of the inclosed air; but this may be pre
vented, if needful, by introducing air of a lower than atmospheric pressure at
an ordinary temperature, even so low as onefourth of an atmosphere;
for, although the apparatus is less sensitive in proportion as the first supply
of air is of less density and pressure, yet withal it is sufficiently sensitive.
The thermometer, as employed by M. Regnault, is shown in Fig. 120. Two
glass tubes, df^ cd, about halfaninch bore,
are united at the base by a stopcock r.
The tube cd is open above, and df is con
nected to the reservoir a by a small tube ab.
The cover of the boiler in which the reser
voir is inclosed is shown at b, and the tubes
are protected from the heat of the boiler by
the partition c d. By means of a threeway
connection, g^ and tube hy the connecting
tube ab communicates with an air pump,
by means of which the apparatus may be
dried, and air or other gas supplied to it
The first thing to be done is to completely
dry the apparatus, and for this object, a little
mercury is passed into the tube bd^ and the
cock r is closed against it. The exhausting
pump is then set to work to exhaust the
tube, which is done several times, the air
being slowly readmitted after each exhaus
tion, after having been passed through a
filter of pumicestone in connection with
the pump, saturated with concentrated sul
phuric acid to absorb moisture, and thus
desiccate the air. During this part of the
process, the reservoir is maintained at a
temperature of 130° F., or 140° F., to insure
complete desiccation. Next, the reservoir is plunged into melting ice,
the two vertical tubes bd^ cd, are put into conynunication, and filled with
mercury up to a suitable level /, marked on the tube bd. If it is desired
to establish an internal pressure less than that of the atmosphere, the air
is partially exhausted by means of the pump, the degree of exhaustion being
recorded by the difference of level in the two tubes. The exhausting tube
h is then hermetically sealed, and the mercury adjusted to the level /in the
tube bd.
G
Fig. Z20.
Pyrometers.
Pyrometers are employed to measiue temperatures above the boiling
point of mercury, about 676* F. They depend upon the change of form of
either solid or gaseous bodies, liquids being necessarily inadmissible.
Pyrometric estimations are of three classes : — First, those of which the
PYROMETERS. 32/
indications are based upon the change of dimensions of a particular body,
solid or gaseous — the pyrometer; second, those based on the heat imparted
to water by a heated body; third, those which are based upon the melting
points of metals and metallic alloys.
Wedgwood^s pyrometer, invented in 1782, was founded on the property
possessed by clay of contracting at high temperatures, an eflfect which is
due partly to the dissipation of the water in clay, and subsequently to partial
vitrification. The apparatus consists of a metallic groove, 24 inches long,
the sides of which converge, being halfaninch wide above and threetenths
below. The clay is made up into little cylinders or truncated cones, which
fit the commencement of the groove after having been heated to low red
ness; their subsequent contraction by heat is determined by allowing them
to slide from the top of the groove downwards till they arrive at a part of
it through which they cannot pass. The zero point is fixed at the tempera
ture of low redness, 1077° F. The whole length of the. groove or scale is
divided into 240 degrees, each of which was supposed by Wedgwood
equivalent to 130° F., the other end of the scale being assumed to represent
32,277° F. Wedgwood also assumed that the contraction of the clay was
proportional to the degree of heat to which it might be exposed; but this
assumption is not correct, for a longcontinued moderate heat is found to
cause the same amount of contraction as a more violent heat for a shorter
period. Wedgwood's pyrometer is not employed by scientific men, because
its indications cannot be relied upon for the reason just given, and also
because the contraction of different clays under great. heat is not always the
same.
In Daniell's pyrometer the temperature is measured by the expansion of
a. metal bar inclosed in a blacklead earthenware case, which is drilled out
longitudinally to ^ inch in diameter and 7^ inches deep. A bar of
platinum or soft iron, a little less in diameter, and an inch shorter than the
bore, is placed in it and surmounted by a porcelain index i J^ inches long,
kept in its place by a strap of platinum and an earthenware wedge.
When the instrument is heated, the bar, by its greater rate of expansion
compared with the blacklead, presses forward the index, which is kept in
its new situation by the strap and wedge until the instrument cools, when
the observation can be taken by means of a scale.
The airpyrometer. The principle and construction of the airthermo
meter are directly applicable for pyrometric purposes, substituting a platinum
globe for the glass reservoir already described, for resisting great heat, and
as large as possible. The' chief cause of uncertainty is the expansion of
the metal at high temperatures.
The second means of estimation is best represented by the " pyrometer "
of Mr. Wilson, of St. Helen's. He heats a given weight of platinum in the
fire of which the temperature is to be measured, and plunges it into a
vessel containing twice the weight of water of a known temperature.
Observing the rise of temperature in the water, he calculates the tempera
ture to which the platinum was subjected, in terms of the rise of tempera
ture of the water, the relative weights of the platinum and the water, and
their specific heats. In fact, the elevation of the temperature of the
water is to that of the platinum above the original temperature of the water
in the compound ratio of the weights and specific heats inversely; that is
to say, that the weights of the platinum and the water being as i to 2, and
328 HEAT.
their specific heats as .0314 to i, the rise of temperature of the water is to
that of the platinum as i x .0314 to 2 x i, or as i to 63.7, and the rule for
finding the temperature of the fire is to multiply the rise of temperature of
the water by 63. 7, and add its original temperature to the product. The
sum is the temperature of the fire, subject to correction for the heat
absorbed by the thermometer in the water, and by the iron vessel contain
ing the water, and the heat retained by the platinum. The correction is
estimated by Mr. Wilson at iV^j taking the weight of water at 2000 grains,
and that of the platinum 1000 grains, and it may be allowed for by increas
ing the abovenamed multiplier by rrth, to 67.45.
Mr. Wilson proposed that for general practical purposes a small piece of
Stourbridge clay be substituted for platinum, to lessen the cost of the
apparatus. With a piece of such clay, weighing 200 grains, and 2000
grains of water, he found that the correct multiplier was 46.
The third means of estimation, based on the melting points of metals
and metallic alloys, is applied simply by suspending in the heated medium
a piece of metal or alloy of which the melting point is known, and, if
necessary, two or more pieces of different melting points, so as to ascertain,
according to the pieces which are melted and those which continue in the
solid state, within certain limits of temperature, the heat of the furnace. A
list of melting points of metals and metallic alloys is given in a subsequent
chapter.
Luminosity at High Temperatures.
The luminosity or shades of temperature have been observed by M.
Pouillet by means of an airpyrometer to be as follows : —
Shaob. Tempsraturs, Temperature,
Centigrade. Fahrenheit.
Nascent Red 525° 977°
Dark Red 700 1292
Nascent Cherry Red 800 1472
Cherry Red 900 1652
Bright Cherry Red 1000 1832
Very Deep Orange iioo 2012
Bright Orange. 1200 2192
White 1300 2372
"Sweating" White 1400 2552
Dazzling White 1500 2732
A bright bar of iron, slowly heated in contact with air, assumes the
following tints at annexed temperatures (Claudel) : —
Centigrade. Fahrenheit
1. Cold iron at about 12*' or 54°
2. Yellow at 225 437
3. Orangeat 243 473
4. Red at 265 509
5. Violetat 277 531
6. Indigo at 288 550
7 Blue at 293 559
8. Green at 332 630
9. Oxide Gray (gris cToxyde) at 400 752
MOVEMENTS OF HEAT. 329
MOVEMENTS OF HEAT.
When two bodies in the neighbourhood of each other have unequal
temperatures, there exists between them a transfer of heat from the hotter
of the two to the other. The tendency to an equalization, or towards an
equilibrium, of temperatures in this way is universal, and the passage of
heat takes place in three ways : by radiation, by conduction, and by con
vection or carriage from one place to another by heated currents.
Radiation of Heat from Combustibles.
It is a common assumption that the radiation of heat from combustibles
is relatively very small in comparison with the total quantity of heat
evolved. Holding the hand near the flame of a candle, laterally, the
radiant heat, which is the only heat thus experienced, is much less than the
heat experienced by the hand when held above the flame, which is the heat
by convection of the hot current of air which rises from the flame. But it
is to be noted that, whilst the radiant heat is dissipated all round the flame,
the diameter of the upward current is little more than that of the flame,
and the conveyed heat is therefore concentrated in a small compass.
M. Peclet, by means of a simple apparatus, consisting of a cage suspend
ing the combustible within a hollow cylinder filled with water in an annular
space, ascertained that the proportion of the total 'heat radiated from
clifrerent combustibles was as follows : —
Radiant heat from wood nearly }(.
Do. do. wood charcoal „ J?.
Do. do. oil „ '/j.
These values serve to show that radiation of heat is considerable, and that
flameless carbon radiates much more than flame, though the proportion of
heat radiated from fuels depends very much upon the disposition of the
material and the extent of radiating surface.
With respect to heated bodies, apart from combustibles as such, the
radiation or emission of heat implies the reverse process of absorption, and
the best radiators are likewise the best absorbents of heat. All bodies
possess the property of radiating heat The heat rkys proceed in straight
lines, and the intensity of the heat radiated from any one source of heat
becomes less as the distance from the source of heat increases, in the
inverse ratio of the square of the distance. That is to say, for example,
that at any given distance from the source of radiation, the intensity of the
radiant heat is four times as great as it is at twice the distance, and nine
times as great as it is at three times the distance.
The quantity of heat emitted by radiatioh increases in some proportion
with the difference of temperatures of the radiating body and the surrounding
medium, but more rapidly than the simple proportion for the greater differ
ences; and the quantity of heat, greater or less, emitted by bodies by radiation
under the same circumstances is the measure of their radiating p<nver.
Radiant heat traverses air without heating it.
When a polished body is struck by a ray of heat, it absorbs a part of the
heat and reflects the rest The greater or less proportion of heat absorbed
by the body is the measure of its absorbing power, and the reflected heat is
the measure of its reeding power.
330
HEAT.
When the temperature of a body remains constant it indicates that the
quantity of heat emitted is equal to the quantity of heat absorbed by the
body. The reflecting power of a body is the complement of its absorbing
power; that is to say, that the sum of the absorbing and reflecting powers of
all bodies is the same, which amounts to this, that a ray of heat striking a
body is disposed of by absorption and reflection together, that which is not
absorbed being necessarily reflected.
For example, the radiating power of a body being represented by 90, the
reflecting power is also 90, and the absorbing power is lo, supposing that
Table No. 106. — Comparative Radiating or Absorbent and
Reflecting Powers of Substances.
Substance.
Lamp Black
Water
Carbonate of Lead
Writing Paper
Ivory, Jet, Marble
Isinglass
Ordinary Glass
China Ink
Ice
Gum Lac
Silver Leaf on Glass
Cast Iron, brightly polished
Mercury, about
Wrought Iron, polished
Zinc, polished
Steel, polished
Platinum, a little polished
Do. deposited on Copper . . .
Do. in Sheet
Tin
Brass, cast, dead polished
Do. hammered, dead polished.,
Do. cast, bright polishefl
Do. hammered, bright polished
Copper, varnished
Do. deposited on iron
Do. hammered or cast
Gold, plated
Do. deposited on polished Steel
Silver, hammered, polished bright
Do. cast, polished bright
Powers.
Radiating or
Absorbing.
Reflecting.
100
100
100
98
2
93 to 98
7 to 2
91
9
90
10
85
15
85
15
72
28
27
73
25
75
23
77
23
77
19
Si
17 '
83
24
76
17
83
17
83
15
85
II
89
9
91
7
93
7
93
14
86
7
93
7
93
5
95
3
97
3
97
3
97
MOVEMENTS OF HEAT.
331
the total •quantity of heat which strikes the body is represented by 100.
The reflecting power of soot is sensibly «/7, and its absorbing and radiating
powers are 100.
The absorbing power varies with the nature of the source of heat, with
the condition of the substance, and with the inclination of the direction of
the heat radiated upon the body. That of a metallic surface is so much
the less, and consequently the reflecting power is so much the more, in
proportion as the surface is better polished.
The reflecting power of metals, according to MM. de la Provostaye and
Desains, is practically the same, when the angle of incidence, that is the
angle at which the rays of heat strike the surface, is less than 70® of inclina
tion with the surface; but for greater angles, approaching more nearly to
90°, perpendicular to the surface, it sensibly diminishes.
For example, at angles of from 75 to 80 degrees, the reflecting power is
only 94 per cent, of what it is under the smaller angles of incidence.
The table No. 106 contains the radiating and absorbing powers and the
reflecting powers of various substances. (Leslie^ De la Provostaye and
JDesainSy and Melloni,)
The reflecting power of glass has been found to be the same for heat and
for light
Conduction of Heat. — Conduction is the movement of heat through sub
stances, or from one substance to another in contact with it. The table
No. 107 contains the relative internal conducting power of metals and earths,
according to M. Despretz. A body which conducts heat well is called a
good conductor of heat; if it conducts heat slowly, it is a bad conductor of
heat. Bodies which are finely fibrous, as cotton, wool, eiderdown, wadding,
finely divided charcoal, are the worst conductors of heat. Liquids and
gases are bad conductors; but if suitable provision be made for the free
circulation of fluids they may abstract heat very quickly by contact with
heated surfaces, acting by convection.
Convection of Heat, — Convected or carried heat is that which is trans
ferred from one place to another by a current of liquid or gas : for example,
by the products of combustion in a furnace towards the heating surface
in the flues of a boiler.
Table No. 107. — Relative Internal Conducting Power of Bodies.
Substance.
Relative conducting
power.
t
Substance.
Relative conducting
power.
Gold
1000
981
973
892
749
562
374
Zinc
363
304
180
24
12
II
Platinum ;
Tin
Silver
Lead
Copper
Marble
Brass
Porcelain
Cast Iron
Terra Cotta
Wrought Iron
THE MECHANICAL THEORY OF HEAT. '
Heat and mechanical force are identical and convertible. Independently
of the medium through which heat may be developed into mechanical
action, the same quantity of heat is resolved into the same total quantity
of work. The English unit of heat is that which is required to raise the
temperature of i lb. of water r degree Fahr. If 2 lbs. of water be raised
1 degree, or i lb. be raised 2 degrees in temperature, the expenditure of
f heaCj and to express the
lies between the unit of
le footpound, on the other
; of the numerical relation
s obtained by the following
iptator. Fig. izi, consisting
of a vertical shaft carry
ing a brass paddlewheel,
of which the paddles re
volved between station
ary vanes, which served
to prevent the liquid in
the vessel from being
bodily whirled in the
direction of rotation.
The vessel was filled
with water, and the agi
tator was made to revolve
bymeans of a cord wound
round the upper part of
the shaft, and attached
to a weight which de
scended m front of a
scale, by which the work
done was measured.
found that the heat com
ited to one pounddegree
expended in producing it.
at was capable of raising
jnical equivalent of heat,
;n as 772 footpounds for
nedium, and it yielded die
alent for different thermo
botpounds,
;s(say424)tilogrammetres.
.60 (say 1390) footpounds,
[lish thermal units — about
MECHANICAL THEORY OF HEAT. 333
According to the mechanical theory of heat, in its general form, heat,
mechanical force, electricity, chemical affinity, light, and sound, are but
different manifestations of motion. Dulong and Gay Lussac proved by
their experiments on sound, that the greater the specific heat of a gas, the
more rapid are its atomic vibrations. Elevation of temperature does not
alter the rapidity but increases the length of their vibrations, and in con
sequence produces "expansion " of the body. All gases and vapours are
assumed to consist of numerous small atoms, moving or vibrating in all
directions with great rapidity; but the average velocity of these vibrations
can be estimated when the pressure and weight of any given volume of the
gas is known, pressure being, as explained by Joule, the impact of those
numerous small atoms striking in all directions, and against the sides of the
vessel containing the gas. The greater the number of these atoms, or the
greater their aggregate weight, in a given space, and the higher the velocity,
the greater is the pressure. A double weight of a perfect gas, when con
fined in the same space, and vibrating with the same velocity — that is,
having the same temperature — gives a double pressure; but the same weight
of gas, confined in the same space, will, when the atoms vibrate with a
double velocity, give a quadruple pressure. An increase or decrease of
temperature is simply an increase or decrease of molecular motion. When
the piston in the cylinder yields to the pressure of steam, the atoms will
not rebound from it with the same velocity with which they strike, but will
return after each succeeding blow, with a velocity continually decreasing
as the piston continues to recede, and the length of the vibrations will be
diminished. The motion gained by the piston will be precisely equivalent
to the energy, heat, or molecular motion lost by the atoms of the gas;
and it would be as reasonable to expect one billiard ball to strike and give
motion to another without losing any of its own motion, as to suppose that
the piston of a steamengine can be set in motion without a corresponding
quantity of energy being lost by some other body.
In expanding air spontaneously to a double volume, delivering it, say,
into a vacuous space, it has been proved repeatedly that the air does not
appreciably fall in temperature, no external work being performed; but that,
on the contrary^ if the air at a temperature, say, of 230° F., be expanded
against an opposing pressure or resistance, as against the piston of a cylinder,
giving motion to it and raising a weight or otherwise doing work, the tem
perature will fall nearly 170® F. when the volume is doubled, that is from
230° F. to about 60® F., and, taking the initial pressure at 40 lbs.,
the final pressure would be 15 lbs. per square inch.
When a pound weight of air, in expanding, at any temperature or pressure,
raises 130 lbs. one foot high,' it loses i® F. in temperature; in other
words, this pound of air would lose as much molecular energy as would
equal the energy acquired by a weight of one pound falling through a
height of 130 feet. It must, however, be remarked that but a small portion
of this work — 130 footpounds — can be had as available work, as the heat
which disappears does not depend on the amount of work or duty realized,
but upon the total of the opposing forces, including all resistance from any
external source whatever. When air is compressed the atmosphere descends
and follows the piston, assisting in the operation with its whole weight; and
when air is expanded the motion of the piston is, on the contrary, opposed
by the whole weight of the atmosphere, which is again raised. Although,
334 HEAT.
therefore, in expanding air, the heat which disappears is in proportion to
the total opposing force, it is much in excess of what can be rendered
available; and, commonly, where air is compressed the heat generated is
much greater than that which is due to the work which is required to be
expended in compressing it, the atmosphere assisting in the operation.
Let a pound of water, at a temperature of 212** F., be injected into a
vacuous space or vessel, having 26.36 cubic feet of capacity — the volume
of one pound of saturated steam at that temperature — ^and let it be evapor
ated into such steam, then 893.8 units of heat would be expended in the
process. But if a second pound of water, at 212°, be injected and evapor
ated at the same temperature, under a uniform pressure of 14.7 lbs. per
square inch, being the pressure due to the temperature, the second pound
must dislodge the first, supposing the vessel to be expansible, by repelling
'that pressure; and this involves an amount of labour equal to 55,800 foot
pounds (that is, 14.7 lbs. x 144 square inches x 26.36 cubic feet), and an
additional expenditure of 72.3 units of heat (that is, 55,800 f 772), making
a total, for the second pound, of 965. i units.
Similarly, when 1408 units of heat are expended in raising the tempera
ture of air under a constant pressure, 1000 of these units increase the
velocity of the molecules, or produce a sensible increase of temperature ;
while the remaining 408 units, which disappear as the air expands, are
directly consumed in repelling the external pressure for the expansion of
volume.
Again, if steam be permitted to flow from a boiler into a comparatively
vacuous space without giving motion to another body, the temperature of
the steam entering this space would rise higher than that of the steam in
the boiler. Or, suppose two vessels, side by side, one of them vacuous and
the other filled with air at, say, two atmospheres; if a communication be
opened between them, the pressure becomes the same in both. But the
temperature would fall in one vessel and rise in the other; and although
the air is expanded in this manner to double its first volume, there would
not, on the whole, be any appreciable loss of heat, for if the separate por
tions of air be mixed together, the resulting average temperature of the
whole would be very nearly the same as at first It has been proved
experimentally, corroborative of this statement, that the quantity of heat
required to raise the temperature of a given weight of air, to a given extent,
is the same, irrespective of the density or the volume of the air. Regnault
and Joule found that to raise the temperature of a pound of air, whether
I cubic foot or 10 cubic feet in volume, the same quantity of heat was
expended.
In rising against the force of gravity steam becomes colder, and it par
tially condenses while ascending, in the effort of overcoming the resistance
of gravity. For instance, a column of steam weighing, on a square inch of
base, 250.3 lbs., that is to say, having a pressure of 250.3 lbs. per square
inch, would, at a height of 275,000 feet, be reduced to a pressure of i lb.
per square inch, and, in ascending to this height, the temperature would
fall from 401° to 102° F., while, at the same time, nearly 25 per cent
of the whole vapour would be precipitated in the form of water, unless it
were supplied with additional heat while ascending.
If a body of compressed air be allowed to rush freely into the atmosphere,
the temperature falls in the rapid part of the current, by the conversion of
EXPANSION BY HEAT. 335
heat into motion, but the heat is almost all reproduced when the motion
has quite subsided. From recent experiments, it appears that nearly
similar results are obtained from the emission of steam under pressure.
When water falls through a gaseous atmosphere, its motion is constantly
retarded as it is brought into collision with the particles of that atmosphere,
and by this collision it is partly heated and partly converted into vapour.
If a body of water descends freely through a height of 772 feet, it acquires
from gravity a velocity of 223 feet per second; and, if suddenly brought to
rest when moving with this velocity, it would be violently agitated, and
would be raised one degree of temperature. But suppose a waterwheel,
772 feet in diameter, into the buckets of which the water is quietly dropped;
when the water descends to the foot of the fall, and is delivered gently into
the tailrace, it is not sensibly heated. The greatest amount of work it is
possible to obtain from water falling from a given level to a lower level is
expressible by the weight of water multiplied by the height of the fall.
These illustrative exhibitions of the nature and reciprocal action of heat
and motive power, show that the nature and extent of the change of tem
perature of a gas while expanding depend nearly altogether upon the cir
cumstances under which the change of volume takes place.
EXPANSION BY HEAT.
All bodies are expanded by the application of heat, but in different
degrees. Expansion is measurable in three directions : — Length, breadth,
and thickness; and it may be measured as linear expansion, in one direc
tion; as superficial expansion, in two directions; or as cubical expansion,
in three directions. Linear expansion, or the expansion of length, is that
which will be exposed in the following tables for solids and liquids. The
expansion of gases is measured cubically, by volume.
Superficial expansion, it may be added, is twice the linear expansion, and
cubical expansion is three times the linear expansion. That is to say, the
additional volume by expansion in two direc
, tions, as in length and breadth, is twice the
I additional volume in one direction; and the
: additional volume in three directions is three
I times that in one direction. For example,
j take a solid cube abcdefg\ the expansion in
' one direction ea^ on the face abed, is, say,
equal to that indicated by the dot lines pro
jected from that face, and the volume by
expansion is equal to the extension of the
surface abed thus projected. In each of the
two other directions, da, upwards, and ab,
*• "^ laterally, the volume by. expansion is the same
as that of the expansion on the face abed. Consequently, the total
increase of volume by expansion, as measured cubically, in the three
directions of length, breadth, and thickness, is three times the increase of
volume in one direction singly; and, as measured superficially, in two of
these directions, it is twice the increase of volume in one direction.
336
HEAT.
Table No. io8. — Linear Expansion of Solids by Heat, between
32° AND 212*" F.
METALS.
}
Zinc, sheet
Do., forged
Lead
Zinc 8+1 tin, slightly ham
mered,
White Solder: — tin i ♦ 2 lead.
Tin, grain
Tin
Silver
Speculum metal
Brass
Copper
Gun Metal: — 16 copper + i tin
8 copper + i tin
Yellow Brass : — Rod
Do. Trough form..
Gold:—
Paris standard,' annealed
Do. unannealed
Bismuth
Iron, forged
Do. wire
Steel, rod, 5 feet long
Do. tempered
Do. not tempered
Cast Iron, rod, 5 feet long
Antimony
Palladium
Platinum
Expansion
between m'
and 213* F.
in common
fractions.
1/
■/3s.
■/:
■/
■/
37a
399
V
403
524
753a
V58X
V5a4
Vsaa
Vsa8
V.
/66x
64s
V719
V8x9
Vsia
V874
y8o7
'/901
^/9a3
/xooo
7x167
Expansion
between 32*
and 2x2' F.
in a length
=ioa
length = xoo.
29416
31083
28484
26917
25053
24833
21730
19075
19333
18782
17220
19083
18167
18930
18945
I5153
I5516
I3917
12204
12350
1 1450
12396
10792
moo
10833
1 0000
08570
Expansion
between 32'
and 2X2* F.
in a length'
of xo feet.
inch.
•353
.374
.342
.322
.301
.298
.260
.229
.232
.225
.207
.229
.218
.227
.227
.181
.186
.167
.146
.148
•137
.149
.130
•133
.130
.120
.103
Expansion
for I F. in a
length of TOO
fecL
inch.
.0196
.0207
.0190
.0179
.0167
.0166
.0145
.0127
.0130
.0125
.0115
.0127
.0121
.0126
.0126
.0101
.0103
.00928
.00814
.00823
.00763
.00826
.00719
.00740
.00722
.00667
.00571
From 0° to 300° C.
(32° F. to 572° F.)
Copper...
Iron
Platinum.
/ 0° to 100° C.
( 0° to 300° C.
/ 0° to loo** C.
\ 0° to 300° C.
{
0° to 100° C.
o** to 300'' C.
1/582
V846
V68x
71x03
Vxo89
.17182
.18832
.11821
.14684
.08842
.09183
.206
.226
.142
.176
.106
.III
.0115
.00418
.00788
.00326
.00589
.00204
LINEAR EXPANSION OF SOLIDS BY HEAT.
337
Table No. io8 {continued).
GLASS.
Flint Glass
French Glass, with lead
Glass tube, without lead
Glass of St. Gobain
Barometer tubes (Smeaton).
Glass tube (Roy)
Glass rod, solid (Roy)
Glass (Dulong and Petit) . . .
Do. (o° to 200° C.)
Do. (o'' ta3oo°C.)
Ice
Expansion
between 32*
and 213* F.
in common
fractions.
/1248
Am7
/1090
/zx2a
/"75
/1289
/"37
V1032
V987
Expansion
between 32*
and 2X2° r.
in a length
=ioa
.08117
.08720
.09175
.08909
.08333
.07755
.08083
.08613
.09484
.10108
Expansion
between 32*
and 2X2* F.
in a length
of xo feet.
inch.
.0974
.105
.110
.107
.100
.0931
.0970
.103
.114
.121
sion
Expansi
for X F. in a
length of xco
feet.
inch.
.00541
.00581
.00612
.00594
.00555
.00517
.00539
.00574
.00632
.00674
■0333
STONES.
Granite
Do
Cky^late
Do.
York paving.
Micaceous sandstone.
Do. do.
Do. do.
Do. do.
Do. do.
Do. do.
Carrara marble
Sost do
Stock Brick ,
Initial ,
Temperature.
Final
Temperature.
45^
45
46
46
46
52
52
52
52
45
45
32
32
52
F.
220
100
87
104
95
200
200
150
100
100
260
212
212
260
F.
Expansion
in a length
=xoo.
length = too.
.2916
.0416
.0416
.0693
.1695
.1736
.1041
.0832
.0520
.0416
.1458
.0849
.0568
.2500
Expansion
for X F. in a
length of xoo
feet.
inch.
.0200
.00908
.0122
.0143
.0415
.0141
.00844
.0102
.01300
.00908
.00814
.00566
.00380
.00144
25"
338 HEAT.
Speaking exactly, the cubical expansion is rather less than three times,
and the superficial expansion tather less than twice, the linear expansion ;
for, in fact, the expanded comers of the body are carried out to the full
square figure, and have not the entering angles shown in the figure, and
there is, in this way, a certain overlapping of the strata of expansion at the
ends, sides, and top.
The same kind of demonstration applies to bodies of any other than a
cubical shape.
A hollow body expands by heat to the same extent as if it were a solid
body having the same exterior dimensions.
The rate of expansion of solids from the freezing point to the boiling
point of water, 32° to 212*' R, is sensibly uniform.
The table. No. 108, gives the linear expansion of a number of metals,
and of glass, between the freezing and boiling points;, and of ice for one
degree, and of stones for various intervals of temperature. Authorities: —
Laplace and Lavoisier, Smeaton, Roy, Troughton, Wollaston, Dulong and
Petit, Froment, Rennie.
Zinc is the most expansible of the metals; it expands fully onethird
per cent, or as much as Vs^iSt part of its length, when heated from 32° F.
to 212° F. Iron expands about oneseventh to oneeighth per cent.; and
castiron and platinum about onetenth per cent. The expansion of metals
proceeds at a less rate above the boiling point than below it. Ice expands
at the rate of V36,oooth of its length for one degree Fahrenheit; which, for
180°, would be Vaooth of its length, — greatly more than that of any metal.
Expansion of Liquids.
The measurement of the expansion of liquids by the application of heat
cannot well be taken lineally; that is, as linear expansion, in th6 sense
in which the expansion of solids is observed. For liquids must be con
tained in vessels, which only admit of expansion in one direction, seeing
that the liquid is limited by the bottom and sides of the vessel, which
throw the whole of the expansion or enlargement of volume upwards. The
observations on the expansion of liquids, therefore, though measured in
one direction only, necessarily indicate the cubical expansion or total
enlargement of volume. But, of course, it is easy to reduce the expansion
of a liquid for comparison with the linear expansion of a solid by taking
onethird of the observed measurement.
When the temperature of water at the freezing point, 32° F., is raised,
the water does not at first expand, but, on the contrary, contracts in volume
until the temperature is raised to 39°.! F., which is 7.1 degrees above the
freezing point. This is called "the temperature of maximum density."
From this point water expands as the temperature rises, until, at 46° F., it
regains its initial volume, that is, the volume at 32° F. Thence, it con
tinues to expand until it reaches the boiling point, 212° F., under one
atmosphere. Passing this point upwards, if the pressure be suitably
increased, water continues to expand with a rise of temperature.
The cubical expansion of water when heated from 32® to 212° F. is
.0466; that is, the volume is increased from i at 32° F. to 1.0466 at 212**
F. This expansion is rather more than 4^ per cent, or between '/aiSt and
'/aad part of the volume at 32°. The expansion of water increases in a
EXPANSION OF LIQUIDS BY HEAT.
339
Table No. 109. — Expan^on and Density of Pure Water,
FROM 32** TO 390° F.
(Calculated by means of Rankine's approximate formiila. )
Tempera
ture.
Comparative
Volume.
Comparative
Density.
Density,
or weight of
X cubic foot.
Weight of
X gallon.
Remarkable Temperatures.
Fahr.
Water at 32'
= 1.
Water at 32'
= I.
Pounds.
Pounds.
/
32^
1. 00000
1. 00000
62.418
lO.OIOI
Freezing point.
35
0.99993
1.00007
62.422
10.0103
391
0.99989
1. 000 1 1
62.425
IO.OII2
Point of maximum density.
40
0.99989
1. 0001 1
62.425
IO.OII2
45
0.99993
1.00007
62.422
10.0103

46
1. 00000
1. 00000
62.418
lO.OIOI
f Same volume and density
1 as at the freezing point
50
1. 00015
0.99985
62.409
10.0087
(Weight taken for ordi
\ nary calculations.
52.3
1.00029
0.99971
62.400
10.0072
55
1.00038
0.99961
62.394
10.0063
60
1.00074
0.99926
62.372
10.0053
1
62
I.OOIOI
0.99899
62.355
10.0000
Mean temperature.
65
I.00II9
0.99881
62.344
9.9982
70
I.OOI60
0.99832
62.313
99933
75
1.00239
0.99771
62.275
9.9871
80
1.00299
0.99702
62.232
9.980
85
1.00379
0.99622
62.182
i 9972
90
1.00459
099543
62.133
9.964
95
1.00554
0.99449
62.074
9955
100
1.00639
099365
62.022
9947
1 Temperature of conden
( ser water.
105
1.00739
0.99260
61.960
9937
no
1.00889
0.991 19
61.868
9.922
•
"5
1.00989
0.99021
61.807
9913
•
120
I.01I39
0.98874
61.715
9.897
125
1. 01239
0.98808
61.654
9.887
130
1. 01390
0.98630
61.563
9873
135
IOI539
0.98484
61.472
9859
140
1.01690
098339
61.381
9.844
145
1. 01839
0.98194
61.291
9.829
150
1. 01989
0.98050
61.201
9.815
155
1. 02164
0.97882
61.096
9799
160
1.02340
0.97714
60.991
9.781
165
1.02589
0.97477
60.843
9757
170
1.02690
0.97380
60.783
9748
17s
1.02906
0.97193
60.665
9728
180
1.03 100
0,97006
60.548
9.711
185
1.03300
0.96828 60.430
9.691
340
HEAT.
Table l^o, io<) {continued).
(Calculated by means of Rankine's approximate formula.)
Tempera
ture.
Comparative
Volume.
Comparative
Density.
Density,
or weight of
I cubic foot.
Weight of
z gallon.
Remarkable Temperatures.
Fahr.
Water at 32*
= I.
Water at 33*
= I.
Pounds.
Pounds.
190
1.03500
0.96632
60.314
9.672
195
1.03700
0.96440
60.198
9.654
200
1.03889
0.96256
60.081
9.635
205
1. 0414
0.9602
59.93
9.61I
210
1.0434
0.9584
59.82
9.594
212
1.0444
0.9575
59.76
9584
Boiling point; by formula j
212
1.0466
0.9555
59.64
9.565
r Boiling point; by direct
( measurement
230
1.0529
0.9499
5936
9.520
250
1.0628
0.941 1
58.75
9.422
270
1.0727
0.9323
58.18
9.331
290
1.0838
0.9227
57.59
9.236
( Temperature of steam of
I 50 lbs. effective pres
( sure per square inch.
298
1.0899
0.9175
57.27
9.185
I Temperature of steam of
338
I.III8
0.8994
56.14
•
9.004
< 100 lbs. effective pres
( sure per square inch.
i Temperature of steam of
366
I.I3OI
0.8850
55.29
8.867
< 150 lbs. effective pres
( sure per square inch.
i Temperature of steam of
 205 lbs. effective pres
( sure per square inch.
390
1. 1444
0.8738
54.54
8.747
greater ratio than the temperature. The annexed table No. 109 shows
approximately the cubical expansion, comparative density, and comparative
volume of water for temperatures between 32° and 212° F., calculated by
means of an approximate formula constructed by Professor Rankine as
follows : —
D, nearly = ■? — ° —
^ / + 461 5
00
(I)
500 / + 461
in which 00=62.425 lbs. per cubic foot, the maximum density of water,
and D, = its density at a given temperature / F.
Rule. — To find approximately the density of water at a given temperaturCy
the maximum density being 62.425 lbs, per cubic foot. To the given tempera
ture in Fahrenheit degrees, add 461, and divide the sum by 500. Again,
divide 500 by that sum. Add together the two quotients, and divide
124.85 by the sum. The final quotient is the density nearly.
EXPANSION OF LIQUIDS BY HEAT. 34 1
The results given by this rule are very nearly exact for the lower tempera
tures, but for the higher temperatures they are too great. For 212° F. the
density of water by the rule is 59.76 lbs. per cubic foot, but it is actually
only 59.64 lbs., showing an error of about Vsooth part in excess.
From the table it appears that the density of water at 46° F., or about
8° C, is the same as at the freezing point, 32*^ F., and that the temperature
of maximum density, 39°. i F., or 4° C., lies midway between those tempera
tures. The expansion of water towards and down to the freezing point is
Vgoooth part of the volume at the temperature of maximum density. It
would appear that in thus expanding from 39^.1 F. downwards, the particles
of water enter on a preparatory stage of separation, anticipating the still
further separation which ensues on the conversion of water into the solid
state; for ice is considerably lighter than water and floats on it, and its
density is little more than ninetenths that of water.
In passing upwards from the freezing point towards higher temperatures,
the increase of volume of watef by expansion, in parts of the volume at
the freezing point, is as follows : —
Expansion in
parts of the volume
at 32' F.
at 52°.3 F. corresponding to the weight per cubic foot
(62.4 lbs.) usually taken for ordinary calcu per cent
latio ns .03
at 62° the mean temperature .10
at 1 00° the temperature of condenser water. .64
at 212° the boiling point 4.66
at 298° the temperature of steam of 50 lbs. effective
pressure per square inch 9.0
at 338** the temperature of steam of 100 lbs. effective
pressure per square inch 1 1.2
at 366° the temperature of steam of 150 lbs. effective
pressure per square inch 13.0
at 390° the temperature of steam of 205 lbs. effective
pressure per square inch 14.4
The expanded volume of some liquids from 32*^ to 212° F. is given in
table No. no; that is, the apparent expansion as seen through glass. It
is shown that alcohol and nitric acid are the most expansible, and water
and mercury the least; the former expand oneninth of their initial volume,
and of the latter, water, as already stated, expands ^/aad part, and mercury
Yesth part of their initial volumes respectively. Observations on the
absolute expansion of mercury are added, and they show that whereas the
apparent expansion in glass is 'As^h part, the real expansion is ^/^s^h part
of the initial volume.
No other liquid besides water has a point of maximum density ; that is,
a point higher than the freezing point of the liquid.
342
HEAT.
Table No. no. — Expansion of Liquids by Heat, from 3 2*^ to 212° F.
Apparent Expansion, in Glass.
Liquid.
Alcohol
Nitric Acid
Olive Oil
Linseed Oil
Turpentine
Sulphuric Ether
Hydrochloric Acid (density 1.137)
Sulphuric Acid (density 1.850)
Water saturated with Sea Salt
Water
Mercury
Volume at aia" F.
volume at 3a* F.=i.
I.IIOO
I.IIOO
1.0800
1.0800
1.0700
1.0700
1.0600
1.0600
1.0500
1.0466
I.OI54
Ex]>ansion in
Vulga^ Fractions.
■/,
18
volume at 33* F.=:i.
}
V.
Vm
Vm
V.7
■■',:
•A
'As
ao
Absolute Expansion of Mercury.
Volume at
aia* F.
Mercury, from 32* to 212** F. ( g** to 100" C), Dulpngand Petit, i. 01 80180
Do. from 212° to 392° F. (100* to 200' C), do. 1.0184331
Do. from 392" to 572° F. (200" to 300'* C), do. i. 0188679
Do. from 32** to 212" F. ( o" to 100' C), Regnault, 1.0181530
Expan
sion.
V
555
V53
Expansion of Gases by Heat.
Gases are divisible into two classes — permanent gases and vapours.
Gases for which great pressure and extremely low temperatures are neces
sary to reduce them to the liquid form, are called permanent gases, and
those which exist in the fluid state under ordinary temperatures, are called
vapours.
The influence of heat in expanding a permanent gas maintained under a
constant pressure, is such that, for equal increments of temperature, the
increments of volume by expansion are also equal or very nearly equal ; in
Other words, the gas expands uniformly, or very nearly uniformly, in pro
portion to the rise of temperature.
Again, it has been observed that when the volume of permanent gases is
maintained constant, the pressure increases uniformly, or nearly uniformly,
with an increase of temperature.
A perfect or ideal gas is one which, under a constant pressure, expands
with perfect uniformity in proportion to the rise of temperature; and of which,
also, when confined to a constant volume, the pressure increases with per
fect uniformity in proportion to the rise of temperature.
When the temperature of atmospheric air is raised from 3 2** to 212° F.,
the following are the total increments of volume or of pressure, according
to the treatment, as determined by Regnault, when the volume at 32** is
taken as i : —
EXPANSION OF GASES BY HEAT.
343
Air. Tbmpsraturb. Incrbmsnt.
Pressure constant 32° to 212" F Volume increased from i to 1.3670.
Volume constant 32° to 212° F Pressure increased from i to 1.3665.
Showing that the increase of pressure, .3665, with a constant volume, is
sensibly the same as, though less than, the expansion or increase of volume,
.3670, when the pressure is constant
The table No. 1 1 1 gives the expansion and the increase of pressure, for
several gases, when raised from 32° to 212'' F.: —
Table No. in. — Expansion and Pressure of Gases raised from
32°t0 2I2'*F.
(Renault.)
Gasbs.
Expannon of Gases under x Atmosphere.
Increase of Pressure
of Gases under a Con
tant Volume.
Final Volimic at aia' F.
Estpansion at sza* F.,
in Common Fractions.
Fizud Pressure at 2x3*.
Atmospheric Air
Hvdroflfen
Initial volume at 32*=!.
1.3670
1. 3661
1.3669
I.3710
I.3719
13877
13903
Initial volume at 39*= z.
/a.73
;A.73
/2.71
/a. 7a
Va.6i
Va.6o
Initial pressure at 32*= X.
1.3665
1.3667
1.3668
1.3667
1.3688
1.3676
1.3829
1.3843
Nitrocren
Carbonic Oxide
Carbonic Acid
Nitrous Oxide
Cvanocen
Sulphurous Acid
Table No. 112. — Expansion of Gases raised from 32° to 212° F.,
under Different Pressures, these Pressures remaining Con
stant FOR EACH Observation.
(Regnault.)
Gas.
Pressure.
Volume at 2x2*.
Air
•
Millimetres.
760
2525
2620
Atmospheres.
I. GO
3.32
3.45
Volume at 32' F. = x.
1.36706
1.36944
1.36964
Hydrogen
760
2545
1. 00
3.35
I.36613
I.36616
Carbonic Acid
760
2520
1. 00
332
1.37099
1.38455
Sulphurous Acid
760
980
1. 00
1. 16
1.3903
1.3980
344 HEAT.
The first part of the table, No. iii, on the expansion of gases by heat,
shows that the expansion, which is a little more than a third of the initial
volume, is nearly die same for air, hydrogen, and carbonic oxide, which are
sensibly perfect gases, and have never been liquefied. On the contrary,
carbonic acid, cyanogen, and sulphurous acid have a greater enlargement of
volume than those gases, and they are gases which may easily be liquefied.
The second part of the table, column 4, shows that, when the volume
is constant, the pressure is increased nearly in the same proportion as the
volume is increased, when the pressure is constant. This nearness of the
proportions is particularly close in the cases of the three sensibly perfect
gases, — ^air, hydrogen, and carbonic oxide.
The next table. No. 112, contains the results of Regnault's experiments
on the expansion of gases from 32° to 212° F., under various constant
pressures of from i to 3^ atmospheres. It is shown that the expansions of
air and of hydrogen are sensibly the same, whether the constant pressure be
I atmosphere or between 3 and 4 atmospheres ; whilst the expansions of
carbonic acid and sulphurous acid are higher at the higher pressure.
The deductions of Regnault, from his experiments, comprised the
following principles : —
That for air, and all other gases except hydrogen, the coeffifcient of
dilatation, or the increment of expansion for one degree rise of temperature,
increases to some extent with their density.
That all gases possess the same coefficient of dilatation when in a state
of extreme tenuity ; but that this law is departed from as gases become
dense.
Adopting, nevertheless, the mean of the results of the experiments of M.
Regnault and of M. Rudberg, the expansion of one volume of air measured
at 32** F., when heated to 212° F., under a constant pressure, will, for future
calculation, be taken as equal to 0.365; the ratio of the initial to the
expanded volume being as i to 1.365. As the expansion is uniform with
the rise of the temperature through 180°, the expansion for each degree
Fahr. is —
.365^180 = —^  ,
4932
the volume at 32° F. being = i. The same uniform rate of expansion holds
sensibly for temperatures higher than 212°; it has been verified experi
mentally up to 700° F., under one atmosphere. It is inferred that, con
versely, air would contract uniformly under uniform reductions of temperature
below 32^* F., until, on arriving at 493°. 2 below the freezing point, or
46 1**. 2 F. below zero, the air would be reduced to a state of collapse,
without elasticity. This point in the Fahrenheit scale has thus been
adopted as that of absolute zero, standing at the foot of the natural scale of
temperature; and the temperature, measured from absolute zero, or
— 461^2 F., is called the absolute temperature.
Accordingly, if a given weight of air at 0° F. be raised in temperature to
+ 461° F., under a constant pressure, it is expanded to twice its original
volume; and if heated from 0° F. to twice 461°, or 922°, its original
volume is trebled.
In briefi ^^ follows that, sensibly.
EXPANSION OF GASES BY HEAT. 345
ist The pressure of air varies inversely as the volume when the tempera
ture is constant
2d. The pressure varies directly as the absolute temperature when the
volume is constant.
3d. The volume varies as the absolute temperature when the pressure is
constant.
4th. The product of the pressure and volume is proportional to the
absolute temperature.
The absolute zeropoint by different thermometrical scales is as follows : —
Reaumur  2 1 9°. 2
Centigrade 274°
Fahrenheit  46i°.2
To simplify calculation, the decimal is usually dropped from the Fahrenheit
temperature, which is taken as 461°.
The foregoing laws do not apply exactly to the expansion and contraction
of the more easily condensable gases, for these, as they approach the point
of liquefaction, become sensibly more compressible than air. Oxygen,
nitrogen, hydrogen, nitric oxide, and carbonic oxide follow the same ratio of
compression as that of air, being incondensable gases, at least as far as 100
atmospheres of pressure. Sulphurous acid, ammoniacal gas, carbonic acid,
and protoxide of nitrogen, which have been proved, on the contrary, to be
condensable, become sensibly more compressible than air when they are
reduced to onethird or onefourth of their original volume at atmospheric
pressure. Carbonic acid, under five atmospheres, occupies only 97 per
cent of the volume which air occupies under the same pressure ; and under
forty atmospheres, near the condensing point, it occupies only 74 per cent,
or barely threefourths of the volume of air at the same pressure. It has,
nevertheless, been established that all gases, at some distance from the
point of maximum density for the pressure, beyond which point they must
condense, sensibly follow the first law above recited, according to which the
pressure and the density vary directly as each other, when the temperature
is constant With such limitations, they rank as perfect gases.
The table No. 113 contains examples of the progressive pressures required
to compress air, nitrogen, carbonic acid, and hydrogen, into onetwentieth
of their original volumes, founded on experiments made by M. Regnault
The pressures are expressed in metres of mercury, the pressure of a column
of mercury one metre high being equal to 19.34 lbs. per square inch. The
table shows that hydrogen is the most perfect type of gaseity. When
compressed to a twentieth of its original volume, it supports something
more than twenty times the original pressure. Air, on the contrary, requires
a quarter of a metre less than 20 metres of pressure ; nitrogen requires a
fifth of a metre less; and carbonic acid, like an overloaded spring, 3^
metres less.
346
HEAT.
Table No. 113. — Compression of Gases by Pressure under a
Constant Temperature.
1
Ratio of the
Pressiuv in Metres of Mercury for
original volume
to the reduced
volume.
Air.
Nitrogen.
Carbonic Acid.
Hydrogen.
Metres.
Metres.
Metres.
Metres.
I
1.000
1. 000
1. 000
1. 000
2
1.998
1.997
1.983
2.001
4
3987
3.992
3.897
4.007
6
5970
5.980
5743
6.018
8
7.946
7.964
7519
8.034
10
9.916
9944
9.226
10.056
12
11.882
II.919
10.863
12.084
14
13845
13.891
12.430
I4.II9
16
15.804
15.860
13.926
16.162
18
17763
17.825
15.351
18.2II
20
19.720
19.789
16.705
20.269
Note, — 20 metres of mercury are equal to a pressure of 386.8 lbs. per square inch, or
26.3 atmospheres.
Relations of the Pressure, Volume, and Temperature of Air
and other Gases.
In accordance with the relations of pressure, volume, and temperature
above stated, it is found that air and other perfect gases, and, within
practical limits, the permanent gases generally, are expanded by heat at the
rate of V461 part of their volume at 0° F. for each degree of temperature,
under a constant pressure. If the volume at the freezing point, 32° F., be
taken as the point of departure, the denominator of the fraction is
461° + 32° = 493°, and the expansion is at the rate of ^453 part of the
volume at 32° F. for each degree of temperature. In general, for any
other initial temperature the denominator of the fraction showing the rate
of expansion for each degree is found by adding 461° to the initial tempera
ture. But, for convenience of calculation, the initial temperature is usually
taken at 0° F.
Similarly, the pressure of air having a given constant volume, is increased
by heat at the rate of 7461 part of the pressure at 0° F.
The fraction of expansion when the pressure is constant, and the fraction
of pressure when the volume is constant, for each degree of temperature by
Fahrenheit's scale above o^, is, then.
461'
and the same fraction expresses the rate of contraction of volume for each
degree of temperature below o® F.
A number of proportions and rules for the relations of the pressure,
volume, and temperature of a constant weight of a gas are readily deduced
from the above defined ratios.
PRESSURE, ETC., OF AIR AND OTHER GASES. 347
1. When the pressure is constant, the volume varies as the absolute
temperature; or,
V : V: : / + 461 : /' + 461, and
v/_v ^ + 461 . / \
" 7T^' ^^
in which V is the volume of the air or other gas at the temperature /, and
V is the volume at the temperature /^. Whence the rule —
Rule i. To fi?id the volume of a constant weight of air or otJier permanent
gas, at any other temperature, when the volum^ at a given temperature
is known, the pressure being constant. Multiply the given volume by the
new absolute temperature, and divide by the given absolute temperature.
The quotient is the new volume.
Note, — The absolute temperature is found by adding 461° to the
temperature indicated by the Fahrenheit thermometer.
A common case of the above rule is, air at the mean temperature, 62** F.,
and mean atmospheric pressure, 14.7 lbs. per square inch. The increased
volume, by expansion by heat, taking the initial volume = i, is found by
substitution and reduction to be as follows : —
v'jy^^2 (2)
523
Rule 2. To find the increased volume of a constant weight of air, of
which the initial volume =1, taken at 62° F., heated under the ordinary
atmospheric pressure of 14.7 lbs. per square inch, to a given temperature.
To the given temperature add 461, and divide the sum by 523. The
quotient is the increased volume by expansion.
2. When the temperature of a constant weight of air, or other gas, is
constant, the volume varies inversely as the pressure ; or,
V : V : : /' : /, and
V' = V^; (3)
in which V and V are the volumes respectively at the pressures/ and/'.
Rule 3. To find the volume of a constant weight of air or other permanent
gas, for any pressure, when the volume at a given pressure is known,
the temperature remaining constant. Multiply the given volume by the
given pressure, and divide by the new pressure. The quotient is the new
volume.
3. When the pressure and temperature of a constant weight of air or
other gas both change, the volume varies in the compound ratio of the
absolute temperature directly, and the pressure inversely ; or,
V : V ::/(/ + 46i):/(/' + 46i);
or V'/ (/ + 46i) = V/ (/'+461), and
V/.v /(^ + 46i) , V
34^ HEAT.
Rule 4. To find the volume of a coftstant weight of air or other permanait
gas for any other pressure and temperature, when the volume is known
at a given pressure and temperature. Multiply the given volume by the
given pressure, and by the new absolute temperature, and divide by the
new pressure, and by the given absolute temperature. The quotient is the
new volume.
4. When the volume and temperature of a constant weight of air or
other gas both change, the pressure varies in the compound ratio of the
absolute temperature directly, and the volume inversely.
/ :/ : : V'(/ + 46i) : V(/' + 46i);
or V'/(/ + 46i) = V/ (/' + 461), and
^ ^V'(/+46i) ^^^
Rule 5. To find the pressure of a constant weight of air or other permanent
gas for any other volume and temperature, when the pressure is known for
a given volume and temperature. Multiply the given pressure by the
given volume, and by the new absolute temperature, and divide by the
new volume, and by the given absolute temperature. The quotient is the
new pressure.
For the common case, when the initial temperature is 62® F., and the
initial pressure is 14.7 lbs. per square inch, the formula (5) becomes, by
substitution and reduction,
v=VJ^+_46xi (6)
^ 3558 V' ^ >
Rule 6. To find the pressure of a constant weight of air or other gas taken
at 62° F., and at 14.7 ibs, pressure per square inch, with a given volume, for
any other volume and temperature. Multiply the initial volume by the
final temperature plus 461, and divide the product by the final volume,
and by 35.58. The quotient is the new pressure in lbs. per square inch.
When the volume is constant, with an initial temperature of 62** F., and
an initial pressure of 14.7 lbs. per square inch, the above formula (6) is
simplified thus : —
p'=(±^ ^ J
^ 3558 ^^'
Rule 7. To find the pressure of a constant weight of air or other gas takefi
at 62° F., and at 14.7 lbs. pressure per square inch, with a constant volume,
for a given temperjiture. Add 461 to the given temperature, and divide
the sum by 35.58. The quotient is the pressure in lbs. per square inch.
5. The mutual relations of pressure, volume, and temperature are con
densed in the following formula : —
V/*/ + 46i, {a)
the product of the volume and pressure of a constant weight of air being
proportional to the absolute temperature. And, as that product bears
always the same ratio to the absolute temperature, an equation may be
PRESSURE, ETC., OF AIR AND OTHER GASES.
349
formed between them by multiplying the absolute temperature by a
coefficient, which may be put = a. Then —
V/ = ^ (/+461); {b)
that is, the product of the volume and pressure of a constant weight of air
or other permanent gas, is equal to the absolute temperature multiplied by
a constant coefficient, which is to be determined for each gas according to
its density.
Special Rules for One Found Weight of a Gas.
The application of formula {b) to a particular constant weight of gas, will
suffice for many purposes. Let the constant weight be one pound of gas.
To settle the coefficients for the different gases, take, for example, the
temperature 32° F., giving an absolute temperature of 493**, and the pressure
one atmosphere, or 14.7 lbs. per square inch. The volume of one pound
of air at this temperature and this pressure is as before stated, 12.387 cubic
feet Substitute these values for V, /, /, in the formula (^), then —
12.387 X 14.7 =ax 493,
whence the coefficient, a, for air is —
^ = 36935* or
2.7074
and the formula {b) becomes, for air,
/ + 461.
V/ =
2.7074
(O
Table No. 114. — Of Coefficients or Constants, «, in the Equation
{b) FOR THE Relations of the Volume, Pressure, and Tem
perature OF Gases; namely, ^ p  a (/ + 461).
>
Name of gas.
Volume of one pound of
gas, at 32** F., under
one atmosphere.
Value of coefficient a.
Hydrogen
Gaseous steam
cubic feet.
178.83
19913
12.753
12.580
12.387
11.205
8.157
8.IOI
4.777
1.776
533200, or 7o.x875
0.59372, or 7^.6842
0.38027, or 72.6297
0.37506* or 7a.6662
0.36935* or 72.7074
0.33406, or 7a.5535
0.24322, or 74.„x4
0.24155, or 74.,3p9
0.14246, or 77.019s
0.05296, or 7,8.878
Nitrogen
defiant gas
Air
Oxygen
Carbonic acid (ideal)* * .
Do. do. (actual)
Ether vapour*
Vapour of mercury*
* The densities are computed by Rankine for the ideal condition of perfect gas.
3SO HEAT.
that is to say, the volume of one pound of air, multiplied by the pressure
per square inch, is equal to the absolute temperature divided by the
constant 2.7074.
To adapt the formula (d) for other gases, the respective coeflScients, or
constants, are found in the same manner, in terms of the volume of one
pound of each gas, at 32° F., under one atmosphere of 14.7 lbs. per square
inch. They are given in table No. 114.
6. The volume of one pound of air at any pressure and temperature is
deduced as follows : —
V=^+46i ^gj
2.7074/ ^ '
Rule 8. — To find the volume of ofie pound of air, of a given temperature
and pressure. Divide the absolute temperature by the pressure in lbs. per
square inch, and by 2.7074. The quotient is the volume in cubic feet.
For the ordinary case When the pressure is constant at 14.7 lbs, per
square inch, the formula (8) becomes, by substituting and reducing,
^ = lPo ^9)
Rule 9. — To find the volume of one pound of air under 14.7 lbs, pressure
per square inch, at a given temperature. Add 46 1 to the temperature, and
divide the sum by 39.80. The quotient is the volume in cubic feet.
7. The pressiure of one pound of air of any volume, and at any tempera
ture, is found as follows : —
^ = ^7^4V <'°)
Rule 10. — To find tJie pressure of one pound of air, of a given temperature
and volume. Divide the absolute temperature by the volume and by 2.7074.
The quotient is the pressure in lbs. per square inch.
8. The temperature of one pound of air of any volume and pressure is
found as follows : —
/= 2.7074 V/ 461 (11)
Rule i i. — To find the temperature of one pound of air, of a given relume
and pressure. Multiply the volume by the pressure in pounds per square
inch, and also by 2.7074; subtract 461 from the product. The remainder
is the temperature.
9. The density of air is inversely as the volume, and is expressed by an
inversion of the formula (8), for the volume; thus, putting D for the density,
or the weight in pounds of one cubic foot of air —
^ = '7°747:j^ (")
Rule 12. — To find the density of air, at a given temperature and pressure.
Multiply the pressure in pounds per square inch by 2.7074, and divide
by the absolute temperature. The quotient is the density, or weight in
pounds of one cubic foot.
VOLUME, DENSITY, AND PRESSURE OF AIR.
351
Table No. 115. — Volume, Density^ and Pressure of Air at various
Temperatures.
1
1
Volame of one pound of air at
Density, or weight
Pressure of a given weight of air
having a constant volume.
Temperature.
constant atmospheric pressure.
of one cubic foot of
14.7 lbs. per square inch.
Daium— Volume at 62"' F. = i.
air at atmospheric
Datum — Atmospheric pressure at
62 f: = X.
pressure.
Fahrenheit.
cubic feet.
comparative
volume.
pounds.
pounds per
square inch.
comparative
pressure.
0^
.11583
.881
.086331
12.96
.881
32
12.387
•943
.080728
13.86
943
40
12.586
958
.079439
14.08
958
50
12.840
977
.077884
14.36
977
62
13141
1. 000
.076097
14.70
1. 000
70
13342
1.015
.074950
14.92
1.015
80
13.593
1.034
.073565
15.21
1.034
90
13845
1054
.072230
1549
1.054
ICO
14.096
1.073
.070942
15.77
1073
120
14592
I. Ill
.068500
16.33
I. Ill
140
15.100
1.149
.066221
16.89
1.149
160
15603
1. 187
.064088
17.50
1. 187
180
16.106
1.226
.062090
18.02
1.226
200
16.606
1.264
.060210
18.58
1.264
210
16.860
1.283
059313
18.86
1.283
212
16.910
1.287
.059135
18.92
1.287
220
I7.III
1.302
.058442
19.14
1.302
230
17.362
1.321
057596
19.42
1. 321
240
17.612
I 340
.056774
19.70
1.340
250
17.865
1359
.055975
19.98
1359
260
18.I16
1379
.055200
20.27
1379
270
18.367
1.398
054444
20.55
1.398
280
18.621
1.417
053710
20.83
1.417
290
18.870
1.436
.052994
21. II
1.436
300
I9.I2I
1455
.052297
21.39
1455
320
19.624
1493
.050959
21.95
1493
340
20.126
1532
.049686
22.51
1532
360
20.630
1570
.048476
23.08
1.570
380
21. 131
1.608
.047323
2364
1.608
400
21.634
1.646
.046223
24.20
1.646
425
22.262
1.694
.044920
24.90
1.694
450
22.890
1.742
.043686
25.61
1.742
475
23518
1.789
.042520
26.31
1.789
500
24.146
1.837
.041414
27.01
1837
525
24.775
i.«85
.040364
27.71
•1.885
550
25403
1933
.039365
28.42
1.933
575
26.031
1.981
.038415
29.12
1.981
600
26.659
2.029
.037510
29.82
2.029
352
HEAT.
•
Table No.
115 {continued).
Volume of one pound of air at
Density, or weight
Pressure of a given weight of air
having a constant volume.
constant atmospheric pressure.
of one cubic foot of
Temperature.
147 lbs. per square inch.
Datum — ^Volume at 62" F. = i.
air at atmospheric
Datum — Atmosc
6a' F.
thenc pressure at
pressure.
= X.
Fahrenheit.
cubic feet.
comparative
volume.
pounds.
pounds per
square inch.
comparative
pressure.
650
27.915
2.124
.035822
31.23
2.124
700
29.172
2.220
.034280
32.63
2.220
750
30.428
2.315
.032865
34.04
2.315
800
31.685
2.41 1
.031561
35.44
2.41 1
850
32.941
2.507
.030358
36.85
2.507
900
34.197
2.602
.029242
38.25
2.602
950
35.453
2.698
.028206
39.66
2.698
1000
36.710
2.793
.027241
41.06
2.793
1500
49.274
3.749
.020295
55.12
3.749
2000
61.836
4.705
.016172
69.17
4.705
2500
74.400
5.661
.013441
83.22
5.661
3000
86.962
6.618
.011499
97.28
6.618
Note to Rules 8, 9, 10, 11, 12. — ^The coefficients or constants for other
gases, in the application of the preceding five formulas and rules, are given
in table No. 114.
The table No. 115 contains the volume, density, and pressure of air at
various temperatures from 0° to 3000° F., starting from 62° F. and 14.7 lbs.
per square inch respectively as unity for the proportional volumes and pres
sures. The second column of the table, containing the volumes of one pound
of air at different temperatures, was calculated by means of the formula (9),
page 350. The third column, of comparative volumes, the volume at 62°
F. being = i, was calculated by means of formula (2), page 347. The fourth
column, of density, contains the reciprocals of the volumes in column 2,
but it is calculable independently by means of formula (12), page 350. The
fifth column, of pressures, due to the temperatures, was calculated by means
of formula (7), page 348. The sixth column contains the pressures expressed
comparatively, the atmospheric pressure, 14.7 lbs. per square inch, being
taken as i.
SPECIFIC HEAT.
The specific heat of a body signifies its capacity for heat, or the quantity
of heat required to raise the temperature of the body one degree Fahrenheit,
compared with that required to raise the temperature of a quantity of
water of equal weight one degree. The British unit of heat is that which
is required to raise the temperature of one pound of water one degree, from
32° F. to 33** R; and the specific heat of any other body is expressed by
the quantity of heat, in units, necessary to raise the temperature of one
pound weight of such body one degree.
The specific heat of water is represented by i, or unity, and there are
very few bodies of which the specific heat equals or exceeds that of water.
Specific heats are, therefore, almost universally expressible by firactions
of a unit.
SPECIFIC HEAT OF WATER. 353
It is necessary to fix a standard of temperature, such as the freezing
point, for the datum of specific heat, as the specific heat of water is not
exactly the same at different parts of the scale of temperatures, but increases
in an appreciable degree, as well as in an increasing ratio, as the tempera
ture rises. For temperatures not higher than 80° or 90° F., the quantity of
heat required to raise the temperature of water one degree is sensibly
constant ; at 86° F., it is not above onefifth per cent, in excess of that at
the freezingpoint. At 212° F., it is about \y^ per cent, in excess of that
at 32° F. Above 212° F., it increases more rapidly; at 302% it is 2^ per
cent, more than at 32°, and at 402°, it is 4^^ per cent. more.
The average specific heat of water between the freezing and the boiling
points is 1.005, or onehalf per cent, more than the specific heat at the
freezing point.
It follows from the increasing specific heat of water, as the temperature
rises, that the consumption of heat in raising the temperature is slightly
greater expressed in units than in degrees of temperature. To raise, for
example, one pound of water from 0° to 100° C, or from 32** to 212** F.,
there are required 100.5 ^ units, or 180.9 ^' units, of heat.
The specific heats of water in the solid, liquid, and gaseous state are
grouped as follows : —
Ice 0.504
Water i . 000
Gaseous Steam 0.622
showing that in the solid state, as ice, the specific heat of water is only half
that of liquid water ; and that, in the gaseous state, it is a little more than
that of ice, or barely fiveeighths of that of liquid water.
The specific heat of all liquid and solid substances is variable, increasing
sensibly as the temperature rises, and the specific heats of such bodies, as
tabulated, are not to be taken as exact for all temperatures, but rather as
approximate average values, sufficiently near for practical purposes. The
specific heat of the same body is, however, nearly constant for temperatures
under 212° F.
The specific heats of such gases, on the contrary, as are perfectly gaseous,
or nearly so, do not sensibly vary with density or with temperature.
For the same body, the specific heat is greater in the liquid than in the
solid state. For example : —
Liquid. Solid.
Water (specific heat) i .000 o. 504
Bromine „ o.iir 0.084
Mercury „ 0.0333 0.0319
M. Regnault has verified, by numerous experiments, the conclusion
arrived at by previous experimentalists, that, for metals, the specific heats
are in the inverse ratio of their chemical equivalents. Consequently the
products of the specific heats of metals, by their respective chemical
equivalents, are a constant quantity. The same rule holds good for other
groups of bodies of the same composition, and of similar chemical constitu
tion. The specific heat of alloys is sensibly equal to the mean of those of
the alloyed metals.
The following are the specific heats of water for various tempera
88
thermometer, calculated
. (i)
tures from o° to 230° C, or 32° to 446° F., by the a
by means of Regiiault's fonnula : —
c I + 0.00004 /+ 0.0000009/;
in which ^ is the specific heat of water at any temperature /, the specific
heat at the freezing point being  .1.
Table No. 116, — Specific Heat of Water.
01 Specific 1
Units of H«t
fie Hon Bl He
romihefnieiLng , I
e given thefr.
poinl 10 I he giv
prrature. anc
KIHUIK.
F^n=„b.i..
Cecil, unki.
Fahr.uml^ F™
ngpoLnI=..
3!°
0,000
0.000 I
0000
^t
10.002
18.004 '
0005 1
0002
68
aO.oio
36.018
OD12 I
0005
86
30.026
54.047 1
0020 I
0009
104
40,051
72.090 1
0030 I
0013
122
50.087
90.157 I
0042 1
0017
140
60.137
108.247 I
0056
0023
IS«
70.210
126.378 1
0072 1
0030
176
80.282
144.508 1
0089 1
0035
194
90.381
162.686 I
0109 1
0042
ai2
100.500
180.900 I
0130 1
0050
230
110.64.
199.152 I
0153 I
0058
248
120.806
217.449 I
0177 1
0067
266
130.997
235791 1
0204 I
0076
284
141.21S
254187 I
0232 1
0087
302
151.462
272.628 1
0262 I
0097
320
161.741
291.132 1
0294 I
0109
338
172.052
309.690 I
0328 I
356
182.398
328.320 I
0364 1
0133
374
192.779
347.004 1
0401 1
0146
39*
203,200
365.760 I
0440 1
0160
410
213.660
384588
0481 I
0174
428
224.162
403.488 1
0524 1
0.89
446
234.708
422,478 I
0568 1
0204
The Specific Heat of Air and other Gases,
:ific heat, or capacity for heat, of permanent gases is sensibly
r all temperatures, and for all densities. That is to say, the
heat of each g;as is the same for each degree of temperature.
. Regnault proved that the capacity for heat was uniform for
:s varying from 3o°C.to + 225°C. ( 22° to 437° F.); thus
: heat for equal weights of air, at constant pressure, were as
SPECIFIC HEAT OF AIR, ETC. 355
Air between 30"* and + 10° C Specific heat, 0.2377
Do. 10° and+ioo°C Do. 0.2379
Do 100° and + 225"* C Do. 0.2376
Average 0.2377
The temperature is then without any^ sensible influence on the specific
heat of air; neither has the pressure, so far as it has been subjected to
experiment — from one to ten atmospheres — any influence on the magni
tude of the specific heat
The specific heat of gases is to be observed from two points of view: —
I St, When the pressure remains the same, and the gas expands by heat.
' 2d, When the volume remains the same, and the pressure increases with
the temperature. There is a striking diflference in the specific heat, or
capacity for heat, according as it is measured under an increasing volume,
or an increasing pressure. When the temperature is raised one degree,
under constant pressure, with increasing volume, the gas not only becomes
hotter to the same extent as when the volume remains the same and the
pressure alone is increased, but it also expands 7493^ P^"^ ^^ ^^ volume
at 32° F., and thus absorbs an additional quantity of heat in proportion to
the work done by expansion against the pressure. It follpws that the
specific heat of a gas at constant pressure is greater than that of the same
gas under a constant volume; and though the former alone has been made
the subject of direct experiment, the latter being of a diflicult nature for
experimenters, yet the latter, which is properly the specific heat, is easily
deducible from the former on the principle of the mechanical theory of
heat
When the volume of a gas is enlarged by expansion against pressure, the
work thus done in expanding the gas may be expressed in footpounds by
multiplying the enlargement of volume in cubic feet by the resistance to
expansion in pounds per square foot Having thus found the work done
in footpounds, it may be divided by Joule's equivalent, 772, and the
quotient will be the expression of that work in units of heat. It becomes
latent, or insensible to the thermometer, and is called the latent heat of
expansion. It constitutes an expenditure of heat in addition to the heat
that is sensible to the thermometer, and that raises the temperature. The
sum of these two quantities of heat is that which has been observed in the
gross by experimentalists, and which gives the specific heat at constant
pressure.
It follows that, when the specific heat at constant pressure is known, the
specific heat at constant volume may be arrived at by subtracting the pro
portion of heat devoted to the enlargement of the volume from the total
heat absorbed at constant pressure. The remainder is the proportion of
heat necessary and sufficient to elevate the temperature when the volume
remains unaltered, from which the specific heat at constant volume is
deduced by simple proportion; thus —
As the total heat absorbed at constant pressure,
Is to the proportion of heat absorbed at constant volume,
So is the specific heat at constant pressure
To the specific heat at constant volume.
For example, the specific heat of air at constant pressure and with in
I
356 HEAT.
creasing volume has been observed to be .2377, that of water being i. Let
one pound of air at atmospheric pressure, and at 32° F., having a volume
equal to 12.387 cubic feet, be expanded by heat to twice its initial volume,
the pressure remaining the same. The absolute temperature, which is 32°
+ 461 = 493** R, will be doubled, and the indicated temperature will be
32 + 493 = 525° F. Thus, 493 degpees of heat are appropriated, and if the
capacity for heat of the air were the same as that of water, 493 units of heat
would be expended in the process of doubling the volume. But, as the
specific heat is only .2377, or less than a fourth of that of water, the expen
diture of heat is just 493 x. 23 77=117. 18 units, and this quantity comprises
the fraction of heat consumed in displacing the atmosphere and overcoming
its resistance through a space of 12.387 cubic feet additional to the original
or initial volume of the same amount. Now, the work thus done is equal
to—
12.387 cubic feet x 21 16.4 lbs. pressure per sq. foot = 26,216 footpounds;
and dividing this by 772, Joule's equivalent, the work of enlarging or doub
ling the volume is found to be equivalent to 33.96 units of heat Deduct
ing these 33.96 units fi:om the gross expenditure, which is 11 7. 18 units,
the remainder, 83.22 imits, is the proportion of heat required to raise the
temperature through 493 degrees, under an increasing pressure simply,
without increasing the volume ; and this remainder is the datum from which
the proper specific heat of air is to be deduced.
The distribution of heat thus detailed may be concisely exhibited
thus : —
Units.
To double the temperature without adding to the volume.... 83.22
To double the volume, in addition 33'9^
To double the temperature and double the volume at con
stant pressure 117. 18
Now, as before stated, the specific heat at constant volume bears the
same ratio to that at constant pressure, as the respective quantities, or units
of heat, absorbed, do to each other, or as 83.22 and 11 7. 18; and it is found
by simple proportion to be .1688 ; thus —
117. 18 : 83.22 : : .2377 : .1688.
The proper specific heat of air is then .1688, in raising the temperature
without enlarging the volume, and it bears to the socalled specific heat of
air, at constant pressure and with expanding volume, the ratio of i to
1.408.
This ratio, i to 1.408, deduced by means of the mechanical theory of
heat, is practically identical with the ratio experimentally arrived at by M.
Masson from the fall of temperature of a quantity of compressed air, which
was liberated and allowed to expand back until it regained its initial pres
sure. The ratio he deduced firom his inverse experiment was i to 1.41;
which is the ratio of
I to
\/3.
SPECIFIC HEAT OF AIR, ETC. 357
It may be added, by way of explanation, and to enforce the distinction,
that though the pressure of a gas under constant volume rises with the
tem^rature, — a. phenomenon which is analogous, at first sight, to that of the
volume of a gas at constant pressure increasing with the temperature, — yet
there is no expenditure of work in simply raising the pressure in the former
case, while the volume remains unaltered ; whereas, in the latter case, there
is an expenditure in increasing the volume, as has already been shown.
To generalize the foregoing process, by which the specific heat of air at
constant volume has been deduced from the specific heat for constant
pressure; and to show its applicability for finding the specific heat of all
gases at constant volume : —
Given / = the initial temperature of the gas, in degrees Fahrenheit.
„ /' = the final temperature to which the gas is raised.
V = the initial volume of the gas, under one atmosphere of pres
sure, in cubic feet.
V = the final volume of the gas, heated under constant pressure.
„ A = the specific heat of the gas under constant pressure.
Put A' = the specific heat of the gas under constant volume.
H = the total heat expended at constant pressure, in units of heat.
H' = the total heat expended at constant volume.
H^ = the fractional quantity of heat expended in increasing the
volume, at constant pressure; or the latent heat of expansion.
To find the value of A'; then by proportion.
And >4' = ^j.
NowH' = HH^
And — = — ^ — , and, by substitution,
Xx xl
>>
»
>^=<2^' (a)
Again, H = (Z'  /) x ^,
And H^ = {Vt/) X 14.7 X 144^772
= (Vz/)x 2.742;
And^ Jjr^) •
Substituting this value in equation (a) above,
^, >4 (>4 (/'/) 2. 742 (Vz>)) .
* A (/'  /) '
^^^,J{t't)2^i_y{yv) (^)
Whence the following rule : —
Rule i. To find the specific heat of a gas at constant volume, when the
specific heat at constant pressure is given together with the initial and final
temperatures due to given initial and final volumes under an atmosphere of
3S8 HEAT.
pressure. Multiply the difference of the initial and final temperatures by
the specific heat at constant pressure. Also, multiply the difference of the
initial and final volumes by 2.742. Find the difference of these two pro
ducts, and divide it by the difference of the temperatures. The quotient is
ictfic heat of the gas at constant volume.
lying the rule to the example of one pound of air at atmospheric
■e, and at 32° F., doubled in volume by heat; A = .237 7, /'/=493'',
»= 12.387 cubic feet. Then
493
cific heat of air at constant volume, as abeady found,
comparative volumes of other gases are given in table No. 69,
[6, of the Weight and Specific Gravity of G^s and Vapours.
The Specific Heat of Gases for Eajual Volumes.
specific heats of equal volumes of gases are deducible from their
: heats proper, — which are for equal weights. The greater the
, the less is the volume, and the greater the weight of gas that is
iry to equal in volume a lighter gas; it is greater, in fact, in propor
the density.
ce the following rule: —
E 2. To find the specific heat of a gas for equal volumes of the gas
air. Multiply the specific heat of the gas, that is, the specific heat
al weights of the gas and air, by the specific gravity of the gas. The
t is the specific heat for equal volumes.
, — The specific heat for equal volumes may be found for constant
e, and for constant volume, in terms respectively of the specific heat
il weights at constant pressure and constant volume.
ILES OF THE SPECIFIC HeAT OF SOLIDS, LIQUIDS, AND GaSES.
annexed table. No. 117, contains the specific heats of a number of
classified for convenience of reference, into
Metals,
Stones,
Precious Stones,
Sundry Mineral Substances,
Woods.
ipears from the tables that the metals, generally speaking, have the
lecific heat; ranging from bismuth, having a specific heat of .031,
, which has a specific heat of from, .n to .13, and iridium, which
greatest specific heat, namely, .189.
;s show a specific heat of about .20, or a fifth of that of water.
IS stones average less than that.
the sundry mineral substances, glass, sulphur, and phosphorus
^ about a fifth of the specific heat of water, and coal and coke
ds average a half of the specific heat of water.
SPECIFIC HEAT OF SOLIDS.
359
It is a useful practical conclusion, as Dr. Rankine remarks, that the
average specific heat of the nonmetallic materials and contents oif a furnace,
whftier bricks, stones, or fuel, does not greatly diflfer from onefifth of that
of water.
Of the liquids specified in the table No. ii8, it appears that all, with the
exception of bromine, which has a specific heat of i.iii, have less specific
heat than water. Olive oil has the lowest, — only .31 ; alcohol averages .65,
and vinegar, .92.
The table No. 1 1 9 of the specific heat of gases, contains, in the second
column, their specific heat, for equal weights, at constant pressure, as
determined by M. Regnault The third column contains the specific heat,
for equal weights, at constant volume, calculated by means of the Rule i,
above. The fourth and fifth columns contain the specific heat of gases,
for equal volumes, at constant pressure, and at constant volume, arrived
at by means of the Rule 2, above.
There is a remarkable nearness to equality in the specific heat for equal
volumes of air, oxygen, hydrogen, carbonic oxide, and nitrogen. It may
be noted, in particular, that hydrogen, though it has fourteen times the
specific heat of air for equal weights, and has barely a fourteenth of the
density of air, has no more specific heat than air, for equal volumes.
Table No. 117. — Specific Heat of Solids.
(Authority, Regnault^ when not otherwise stated.)
METALS, from 32° to 212° F.
Bismuth ,
Lead
Platinum, sheet
Do. spongy
Do. 32° F. to 212° F {Petit and Duiong)
Do. 32° F. to 572° F. (300° C.) „
Do. at 2i2°F. ( ioo°C.) {PouilUt)
Do. at 572°F. ( 3oo°C.) „
Do. at 932° F. ( 500° C.) „
Do. ati292°F. ( 700° C.) „
Do. at 1832° F. (1000° C.) „
Do. at 2192° F. (1200° C.) „
Gold
Mercury, solid
Do. liquid
Do. 59° to 68° F. (15° to 20° C.)
Do. 32° to 212° F. {Petit and Duiong)
Do. 32° to 572° F. (300° C.)
Tungsten
Antimony
Do. 32° to 212° F {Petit and Duiong)
Do. 32° to 572° F. (300^0.)
Tin, English
Do. Indian
Water at 32°=!.
.03084
•0314
.03243
.03293
.0335
.0355
.0335
.03434
.03518
.036
.03718
.03818
.03244
.0319
.03332
.029
.033
•035
.03636
.05077
.0507
.0547
.05695
.05623
36o
HEAT.
Metals {continued).
Cadmium
Silver
Do. 32** to 212° F {FeHt and Dulong)
Do. 32*^ to 572° F. (300° C.)
Pal ladi um
Uranium
Molybdenum
Brass
Cymbal metal
Copper
Do. 32° to 212° F {Petit and Duiang)
po. 32° to 572° F. (300° C.)
Zinc
Do. 32° to 212° F {Petit and Dulong)
Do. 32°t0 572°F.<3oo°C.)
Cobalt
Do. carburetted
Nickel
Do. carburetted
Wrought iron
Do. 32° to 212° F. {Petit and Dulong
Do. 32? to 392° F. (200° C.) „
Do. 32° to 572° F. (300° C.)
Do. 32° to 662° F. (350° C.) „
Steel, soft
Do. tempered
Do. Haussman
Cast iron, white
Manganese, highly carburetted
Iridium
STONES.
Brickwork and masonry {Rankine) about
Marble, gray
Do. white
Chalk, white
Quicklime
Dolomite (Magnesian limestone)
PRECIOUS STONES.
Sapphire
Zircon ,
Diamond
SUNDRY MINERAL SUBSTANCES.
Tellurium
Iodine
Selenium
Bromine
Phosphorus, 50° to 86° F
Water at 32"=!.
1~
05669'
05701
0557
061 1
05927
0619
07218
0939 ^
086
09515
094
1013
09555
0927
1015
10696
I1714
10863
III92
II379
1098
115
1218
1255
I165
II75
1 1848
12983
14411
1887
20
20989
21585
21485
2169
21743
.21732
•14558
.14687
•05155
.05412
.0837
.0840
.1887
SPECIFIC HEAT OF SOLIDS.
361
Sundry Mineral Substances {continued),
Riosphorus, 32"* to 212° F
Glass
Do. flint
Do. 32° to 212^ F {Petit and Dulong)
Do. 32° to 572° F
Sulphur
Do. crystallized, natural
Do. cast for two years
Do. cast for two months
Do. cast recently
Chloride of lead
Do. zinc
Do. manganese
Do. tin
Do. calcium
Do. potassium
Do. magnesium
Do. sodium
Perchloride of tin
Protochloride of mercury
Nitrate of silver
Do. barytes
Do. potass
Do. soda
Sulphate of lead *.
Do. barytes
Do. potash
Carbonaceous : —
Coal
Charcoal
Coke of cannel coal
Do. pit coal
Coal and coke, average {Rankine)
Anthracite, Welsh
Do. American
Graphite, natural
Do. of blast furnaces
Animal black
Sulphate of lime
Magnesia
Soda
Ice
WOODS.
Turpentine
Pear tree
Oak
Fir
Water at 32*= I.
25034
19768
177
19
20259
1776
1764
1803
1844
06641
13618
14255
14759
16420
17295
19460
.214 to .230
IO161
06889
14352
15228
23875
27821
08723
1 1 285
1901
24111
2415
20307
20085
20
20172
201
20187
497
26085
19659
22159
23115
504
.467
.500
.570
.650
362
HEAT.
Table No. 118. — Specific Heat of Liquids.
>i
yy
Mercury
O live oil (Laplace and Lavoisier)
Sulphuric acid, density 1.87.
Do. do. 1.30.
Benzine, 59° to 68° F
Turpentine,
Do. density .872 (Despretz)
Ether, oxalic
Do., sulphuric, density 0.76 {Dalton)
Do. do. do. 0.715 {Despretz)
Essence of juniper
Do. lemon
Do. orange
Hydrochloric acid
Wood spirit, 59° to 68° F
Chloride of calcium, solution
Acetic acid, concentrated
Alcohol
Do. density o. 793 (Da/ton)
Do. do. 0.81
Vinegar
Water, at 32° F
Do. at 212° F
Do. from 32*
Bromine
»
'°to 212° F.
Water at 32"=!.
.0333
.3096
.3346
.6614
•3932
.4160
.4720
.4554
.6600
.5200
.4770
.4879
.4886
.6000
.6009
.6448
.6581
.6588
.6220
.7000
.9200
1. 0000
1. 0130
1.0050
I. mo
FUSIBILITY OR MELTING POINTS OF SOLIDS.
363
Table No. 119. — Specific Heat of Gases.
Water at 32'' F. = i.
Qas.
Sulphurous acid
Vapour of chloroform
Carbonic acid
Oxygen
Air
Nitrogen
Carbonic oxide
defiant gas
Hydrogen
Vapour of Benzine
Acetic ether
Vapour of alcohol
Gaseous steam ,
Vapour of turpentine . . »
Ammoniacal gas
Light carburetted hydrogen
Specific Heat for
Equal Weights.
At constant
pressure.
water = i.
01553
0.1568
0.2164
0.2182
0.2377
0.2440
0.2479
0.3694
3.4046
03754
0.4008
0.4513
0.4750
0.5061
0.5080
0.5929
At constant
volume.
(Real speci
fic heat.)
water = x.
0.1246
0.1438
O.I 7 14
0.1559
0.1688
o. 1 740
0.1768
0.2992
2.4096
0.3499
0.3781
0.4124
0.3643
0.4915.
0.391 1
0.4683
Specific Heat for
Equal Volumes.
At constant
pressure.
air = ,2377,
as in col. a
0.3489
0.8310
0.3308
0.2412
0.2377
0.2370
0.2399
0.3572
0.2356
I.OII4
1. 2184
O.717I
0.2950
2.3776
0.2994
0.3277
At constant
volume.
air = .1688,
as in col. 3.
0.2799
0.7621
0.2620
0.1723
0.1688
0.1690
O.171I
0.2893
0.1667
0.9427
1. 1490
0.6553
0.2262
2.3090
0.2305
0.2588
FUSIBILITY OR MELTING POINTS OF SOLIDS.
The metals are solid at ordinary temperatures, with the exception of
mercury, which is liquid down to  39° F. Hydrogen, it is believed, is a
metal in a gaseous form.
All the metals are liquid at temperatures more or less elevated, and they
probably vaporize at very high temperatures. Their melting points range
from 39 degrees below zero of Fahrenheit's scale, the melting, or rather the
freezing, point of mercury, up to more than 3000 degrees, beyond the
limits of measurement by any known pyrometer. Certain of the metals,
as potassium, sodium, iron, platinum, become pasty and adhesive at
temperatures much below their melting points. Potassium and sodium,
which melt at temperatures between 130° and 200° F., can be moulded
like wax at 62° F. Two pieces of iron raised to a welding heat, are
softened, and readily unite under the hammer; and pieces of platinum
unite at a white heat.
The melting points of alloys do not follow the ratios of those of their
constituent metals, so that it is impossible to infer their melting points from
these data. A remarkable instance of the absence of this relation is afforded
in the fusible metal consisting of five parts of lead, three of tin, and eight
of bismuth, which melts at 212**^, the heat of boiling water, though the
364
HEAT.
melting point, if it were an average of those of the component metals,
would be about 520'' F. The addition of bismuth to mixtures of lead and
tin has the effect of lowering the melting points.
According to Professor Rankine, the melting point of ice is lowered by
pressure, at the rate of 0.0000063° F. for each pound of pressure on the
square foot. An atmosphere of pressure being 2 116 lbs. per square foot,
the lowering of the melting point per atmosphere of pressure, is —
0°. 0000063 X 2116 = 0° 0133 Fahrenheit.
To lower the melting point one degree, a pressure of 75 atmospheres would
be required.
In the case of water, antimony, and cast iron, and probably other sub
stances, the bulk of the substance in the solid state exceeds that in the
liquid state, as is evidenced by the floating of ice on water, and of solid
iron on molten iron. The volume of water is to that of ice at 32° F., as
I to 1.088; that is to say, that water, in freezing at 32° F., expands nearly
9 per cent.
The following table. No. 120, contains the melting points of metals,
metallic alloys, and other substances : —
Table No. 120. — Melting Points of Solids.
VARIOUS SUBSTANCES {Pauillet, CluMdel, &c)
Sulphurous acid
Carbonic acid
Bromine
Turpentine
Hyponitric acid
Ice
Nitroglycerine
Tallow.....
Phosphorus
Acetic acid
Stearine :
Spermaceti
Margaric acid
Wax, rough
„ bleached
Stearic acid
Iodine
Sulphur
Melting Points.
 148° F.
108
+ 95
14
16
32
45
92
112
"3
109 to 120
120
131 to 140
142
154
158
225
239
MELTING POINTS OF SOLIDS.
365
Table No, 120 (cofiiinued).
METALS.
Mercury
Rubidium
Potassium
Sodium
Lithium
Tin
Cadmium
Bismuth
ThaUium
Lead
Zinc
Antimony
Bronze
Aluminium
Calcium
Silver
Copper
Gold, standard
Gold
Cast Iron, white
i> » gray
„ „ „ 2d melting..
„ „ with manganese..
Steel
Wrought Iron, French
Hammered Iron, English..,.
Malleable Iron
Cobalt
Nickel
Manganese
Palladium
Molybdenum
Tungsten
Chromium
Platinum
Rhodium •.
Iridium
Vanadium
Ruthenium
Osmium
Melting Points.
Pouillet, Claudel.
Fahrenheit degrees.
39°
+ 136
194
446
504
608
680
810
1692
(very pure) 1832
2156
(very pure) 2282
1922 to 2012
2012
2192
2282
2372 to 2552
2732
2912
Wilson.
Fahrenheit degrees.
101°
144
208
356
442
442
617
773
1150
full red heat,
full red heat
1873
1996
2016
2786
Fusible in highest
heat of forge.
Not fusible in forge
fire, but soften and
agglomerate.
Only fusible before
the oxyhydrogen
blowpipe.
366
h£at.
Table No. 120 {continued).
ALLOYS OF LEAD, TIN, AND BISMUTH.
Melting Points.
No.
I.
2.
3
4
5.
6.
7
8.
9.
10.
II.
12.
Tin, 25 Lead.
10
5
3
2
%
2
3
4
5
6
13. 4 Lead, 4 Tin, i Bismuth
^4* 3 » 3 » ^ »
15 2 „ 2 „ I „
16. I „ I „ I „
17. 2 „ I „ 2
18. 3 „ 5 „ 8
HoltzapfTcL
558°
541
482
441
370
334
340
356
365
378
381
320
310
292
254
236
202
ClaudeL
466'
385
367
372
381
SUNDRY ALLOYS OF TIN, LEAD, AND BISMXTTH.
3 Lead, 2 Tin, 5 Bismuth Ure
^ » ' » 2 „ „
1 „ I „ 4 „ Claudd
5 » 3 » 8 „ ; Ure
2 „ 3 » 5 » Claudd
^ » 4 » 5 » >>
' » ' » »
^ >> 3 » »
2 )) I )> ))
I , / Holtzapffel
" ' " t C&w^a'
3 >» ' » >»
3 j» ' >> >»
Melting Points.
199
201
201
212
212
246
286
334
334
360
38s
392
552
ALLOYS FOR FUSIBLE PLUGS.
Softens at
Melts at
2 Tin, 2 Lead
2 „ 6 „
2 » 7 »
2 „ 8 „
365° F.
372
377^4
395 >^
372^ F.
383
388
406 to 410
LATENT HEAT OF FUSION OF SOLID «ODIES.
367
Latent Heat of Fusion of Solid Bodies.
WTien a solid body is exposed to heat, and ultimately passes into